write an essay on mean deviation

Mean Deviation

Mean deviation is used to compute how far the values in a data set are from the center point. Mean, median, and mode all form center points of the data set. In other words, the mean deviation is used to calculate the average of the absolute deviations of the data from the central point. Mean deviation can be calculated for both grouped and ungrouped data.

Mean deviation is a simpler measurement of variability as compared to standard deviation. When we want to find the average deviation from the data's center point, the mean deviation is used. In this article, we will take an in-depth look at mean deviation, its formula, examples as well as the merits and demerits.

What is Mean Deviation?

Mean Deviation falls under average absolute deviation. The average absolute deviation can be defined as the average of the absolute deviations from the central point of the data. The central point can be computed by using either mean, median, or mode .

Mean Deviation Definition

The difference between the observed value of a data point and the expected value is known as deviation in statistics. Thus, mean deviation or mean absolute deviation is the average deviation of a data point from the mean, median, or mode of the data set. Mean deviation can be abbreviated as MAD.

Mean Deviation Example

Suppose we have a set of observations given by {2, 7, 5, 10} and we want to calculate the mean deviation about the mean. We find the mean of the data given by 6. Then we subtract the mean from each value, take the absolute value of each result and add them up to get 10. Finally, we divide this value by the total number of observations (4) to get the mean deviation as 2.5.

Mean Deviation Formula

Depending upon the type of data available as well as the type of the central point, there can be several different formulas to calculate the mean deviation. Given below are the different mean deviation formulas.

Mean Deviation Formula for Ungrouped Data

Data that is not sorted or classified into groups and remains in raw form is known as ungrouped data. To calculate the mean deviation for ungrouped data the formula is as follows:

MAD = \(\frac{\sum_{1}^{n}|x_{i} - \overline{x}|}{n}\)

Here, \(x_{i}\) represents the i th observation, \(\overline{x}\) represents the central point (mean, median, or mode), and 'n' is the number of observations in the data set.

Mean Deviation Formula for Grouped Data

When data is organized and classified into groups it is known as grouped data. Grouping of data is done by continuous and discrete frequency distributions. The mean deviation formulas for grouped data are given below:

Mean Deviation for Continuous Frequency Distribution

Such a type of grouped data consists of class intervals. The frequency of repetition of an observation within each interval is given by the continuous frequency distribution. The mean deviation formula is as follows:

MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-\overline{x}|}{\sum_{1}^{n}f_{i}}\)

\(f_{i}\) is the frequency of repetition of \(x_{i}\). \(x_{i}\) denotes the mid value of the class interval.

Mean Deviation for Discrete Frequency Distribution

In this type of data, the individual data points are specified and the frequency with which they occur is also mentioned. To calculate the mean deviation for a discrete frequency distribution, the formula is given as follows:

\(x_{i}\) denotes the specified individual value and \(f_{i}\) is the frequency of occurrence of that value.

Mean Deviation about Mean

The mean is also known as the expected value of a data set. The simple definition of mean is given as the sum of all observations divided by the total number of observations. The formulas for mean deviation about the mean are given below:

• Ungrouped data MAD = \(\frac{\sum_{1}^{n}|x_{i} - \mu|}{n}\)

where, mean is \(\mu\) = \(\frac{x_{1} + x_{2} +...+x_{n}}{n}\)

• Continuous and discrete frequency distribution MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-\mu|}{\sum_{1}^{n}f_{i}}\)

where, mean of grouped data is \(\mu\) = \(\frac{\sum_{1}^{n}x_{i}f_{i}}{\sum_{1}^{n}f_{i}}\)

Mean Deviation about Median

Median is the value that separates the lower half of the data from the upper half. Given below are the various formulas for the mean deviation about the median:

• Ungrouped data MAD = \(\frac{\sum_{1}^{n}|x_{i} - M|}{n}\)

where, if n is odd , then median M = (\(\frac{n + 1}{2})\) th observation.

if n is even , then median M = \(\frac{\frac{n}{2}^{th}obs + (\frac{n}{2}+1)^{th}obs}{2}\)

• Discrete frequency distribution MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-M|}{\sum_{1}^{n}f_{i}}\)

The median is calculated in the same way as ungrouped data.

• Continuous frequency distribution MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-M|}{\sum_{1}^{n}f_{i}}\)

Median of grouped data M = \(l + \frac{\frac{\sum_{1}^{n}f_{i}}{2}-cf}{f}\times h\)

cf stands for cumulative frequency preceding the median class, l is the lower value of the median class, h is the length of the median class and f is the frequency of the median class.

Mean Deviation about Mode

Mode is defined as that value that occurs most frequently in a given data set. The formulas to calculate mean deviation about mode are as follows:

• Ungrouped data MAD = \(\frac{\sum_{1}^{n}|x_{i} - mode|}{n}\)

where mode = the most frequently occurring value in a data set.

• Discrete frequency distribution MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-mode|}{\sum_{1}^{n}f_{i}}\)

The mode can be calculated in the same way as ungrouped data

• Continuous frequency distribution MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-mode|}{\sum_{1}^{n}f_{i}}\)

where, mode of grouped data = \(l +(\frac{f-f_{1}}{2f - f_{1}-f_{2}})\times h\)

l is the lower value of the modal class, h is the size of the modal class, f is the frequency of the modal class, \(f_{1}\) is the frequency of the class preceding the modal class, and \(f_{2}\) is the frequency of the class succeeding the modal class.

How to Calculate Mean Deviation?

Regardless, of whether the mean deviation about the mean, median or mode needs to be determined, the general steps remain the same. The only difference will be in the formulas used to calculate the mean, median or mode depending upon the type of data available to us. Suppose the mean deviation about the mean has to be determined for the data set {10, 15, 17, 15, 18, 21}. Then the below-given steps can be followed.

• Step 1: Calculate the value of the mean, mode, or median of the given data values. Here, we find the mean given by 16.
• Step 2: Subtract the value of the central point (here, mean) from each data point. (10 - 16), (15 - 16), ..., (21 - 16) = -6, -1, 1, -1, 2, 5.
• Step 3: Now take the absolute of the values obtained in step 2. The values are 6, 1, 1, 1, 2, 5
• Step 4: Take the sum of all the values obtained in step 3. This gives 6 + 1 + 1 + 1 + 2 + 5 = 16
• Step 5: Divide this value by the total number of observations. This results in the mean deviation. As there are 6 observations hence, 16 / 6 = 2.67 which is the mean deviation about the mean.

Mean Deviation and Standard Deviation

Both mean deviation and standard deviation help to measure the variability of data. Given below is the table of differences between mean deviation and standard deviation.

Merits and Demerits of Mean Deviation

Mean deviation is a statistical measure and hence, has its merits and demerits. It is utilized in checking the spread of data with respect to the central value.

Merits of Mean Deviation

Mean deviation is a useful measure as it can remove the shortcomings of other types of statistical measures. Some of the merits are given below:

• It is easy to calculate and simple to understand.
• It does not get extremely affected by outliers.
• It is widely used in business and commerce.
• It has the least sample fluctuations as compared to other statistical measures.
• It is a good comparison measure as it is based on the deviations from the mid-value.

Demerits of Mean Deviation

Mean deviation is not capable of further algebraic treatment hence, this can lead to reduced usability. Other demerits of mean deviation are listed below:

• It is not rigidly defined as it can be calculated with respect to mean, median, and mode.
• Sociological studies rarely use this measure to analyze data.
• Negative and positive signs are ignored because we take the absolute value. This can lead to inaccuracies in the result.

Related Articles:

• Summary Statistics
• Frequency Distributions
• Frequency Distribution Table

Important Notes on Mean Deviation

• Mean deviation is a statistical measure used to give the average value of the absolute deviation with respect to the central point of the data.
• Mean deviation can be calculated about the mean, median, and mode.
• The general formula to calculate the mean deviation for ungrouped data is \(\frac{\sum_{1}^{n}|x_{i} - \overline{x}|}{n}\) and grouped data is \(\frac{\sum_{1}^{n}f_{i}|x_{i}-\overline{x}|}{\sum_{1}^{n}f_{i}}\).
• Mean deviation is less frequently used as compared to standard deviation.

Examples on Mean Deviation

Mean is given by \(\mu\) = \(\frac{\sum_{1}^{5}x_{i}f_{i}}{\sum_{1}^{5}f_{i}}\) = 9.381 Substituting values in the equation, MAD = \(\frac{\sum_{1}^{5}f_{i}|x_{i}-\mu |}{\sum_{1}^{5}f_{i}}\) = 56.378/21 = 2.684 Answer: The mean deviation about mean is 2.684

Answer: Mean deviation about mean = 7.2

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Practice Questions on Mean Deviation

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FAQs on Mean Deviation

What is mean deviation in statistics.

In statistics, the mean deviation is used to give the spread of data about the central point ( mean, median or mode ). It is a type of average absolute deviation.

What is the Formula of Mean Deviation?

The formulas for mean deviation are listed below:

• Ungrouped data MAD = \(\frac{\sum_{1}^{n}|x_{i} - \overline{x}|}{n}\).
• Grouped data MAD = \(\frac{\sum_{1}^{n}f_{i}|x_{i}-\overline{x}|}{\sum_{1}^{n}f_{i}}\).

\(\overline{x}\) denotes either the mean, median or mode depending upon what needs to be computed.

How to Calculate Mean Deviation for Grouped Data?

There can be two types of grouped data - discrete and continuous frequency distributions. The mean deviation for both can be calculated using the formula \(\frac{\sum_{1}^{n}f_{i}|x_{i}-\mu|}{\sum_{1}^{n}f_{i}}\). The mean for such type of data is given by \(\mu\) = \(\frac{\sum_{1}^{n}x_{i}f_{i}}{\sum_{1}^{n}f_{i}}\).

Is Mean Deviation a Measure of Central Tendency?

Mean deviation is not a measure of central tendency. However, it gives the spread of data about the different measures of central tendency such as mean, median, or mode.

Is Mean Deviation a Measure of Dispersion?

Mean deviation is a measure of dispersion. It helps to determine the variability of data with respect to the central value of the given data set.

Is Mean Deviation and Standard Deviation the Same?

No, mean deviation and standard deviation are not the same. Mean deviation calculates the absolute deviations from the central point of the data. However, standard deviation calculates the square of deviations from the mean of the given data.

What are the Advantages and Disadvantages of Mean Deviation?

Mean deviation is very easy to understand and does not get affected greatly by outliers or extreme points in the data. On the other hand, it is not rigidly defined and ignores the negative sign of the data by taking the absolute value.

Chapter 12: Descriptive Statistics

12.3 expressing your results, learning objectives.

• Write out simple descriptive statistics in American Psychological Association (APA) style.
• Interpret and create simple APA-style graphs—including bar graphs, line graphs, and scatterplots.
• Interpret and create simple APA-style tables—including tables of group or condition means and correlation matrixes.

Once you have conducted your descriptive statistical analyses, you will need to present them to others. In this section, we focus on presenting descriptive statistical results in writing, in graphs, and in tables—following American Psychological Association (APA) guidelines for written research reports. These principles can be adapted easily to other presentation formats such as posters and slide show presentations.

Presenting Descriptive Statistics in Writing

When you have a small number of results to report, it is often most efficient to write them out. There are a few important APA style guidelines here. First, statistical results are always presented in the form of numerals rather than words and are usually rounded to two decimal places (e.g., “2.00” rather than “two” or “2”). They can be presented either in the narrative description of the results or parenthetically—much like reference citations. Here are some examples:

The mean age of the participants was 22.43 years with a standard deviation of 2.34. Among the low self-esteem participants, those in a negative mood expressed stronger intentions to have unprotected sex ( M = 4.05, SD = 2.32) than those in a positive mood ( M = 2.15, SD = 2.27). The treatment group had a mean of 23.40 ( SD = 9.33), while the control group had a mean of 20.87 ( SD = 8.45). The test-retest correlation was .96. There was a moderate negative correlation between the alphabetical position of respondents’ last names and their response time ( r = −.27).

Notice that when presented in the narrative, the terms mean and standard deviation are written out, but when presented parenthetically, the symbols M and SD are used instead. Notice also that it is especially important to use parallel construction to express similar or comparable results in similar ways. The third example is much better than the following nonparallel alternative:

Presenting Descriptive Statistics in Graphs

When you have a large number of results to report, you can often do it more clearly and efficiently with a graph. When you prepare graphs for an APA-style research report, there are some general guidelines that you should keep in mind. First, the graph should always add important information rather than repeat information that already appears in the text or in a table. (If a graph presents information more clearly or efficiently, then you should keep the graph and eliminate the text or table.) Second, graphs should be as simple as possible. For example, the Publication Manual discourages the use of color unless it is absolutely necessary (although color can still be an effective element in posters, slide show presentations, or textbooks.) Third, graphs should be interpretable on their own. A reader should be able to understand the basic result based only on the graph and its caption and should not have to refer to the text for an explanation.

There are also several more technical guidelines for graphs that include the following:

• The graph should be slightly wider than it is tall.
• The independent variable should be plotted on the x- axis and the dependent variable on the y- axis.
• Values should increase from left to right on the x- axis and from bottom to top on the y- axis.

Axis Labels and Legends

• Axis labels should be clear and concise and include the units of measurement if they do not appear in the caption.
• Axis labels should be parallel to the axis.
• Legends should appear within the boundaries of the graph.
• Text should be in the same simple font throughout and differ by no more than four points.
• Captions should briefly describe the figure, explain any abbreviations, and include the units of measurement if they do not appear in the axis labels.
• Captions in an APA manuscript should be typed on a separate page that appears at the end of the manuscript. See Chapter 11 “Presenting Your Research” for more information.

As we have seen throughout this book, bar graphs are generally used to present and compare the mean scores for two or more groups or conditions. The bar graph in Figure 12.12 “Sample APA-Style Bar Graph, With Error Bars Representing the Standard Errors, Based on Research by Ollendick and Colleagues” is an APA-style version of Figure 12.5 “Bar Graph Showing Mean Clinician Phobia Ratings for Children in Two Treatment Conditions” . Notice that it conforms to all the guidelines listed. A new element in Figure 12.12 “Sample APA-Style Bar Graph, With Error Bars Representing the Standard Errors, Based on Research by Ollendick and Colleagues” is the smaller vertical bars that extend both upward and downward from the top of each main bar. These are error bars , and they represent the variability in each group or condition. Although they sometimes extend one standard deviation in each direction, they are more likely to extend one standard error in each direction (as in Figure 12.12 “Sample APA-Style Bar Graph, With Error Bars Representing the Standard Errors, Based on Research by Ollendick and Colleagues” ). The standard error is the standard deviation of the group divided by the square root of the sample size of the group. The standard error is used because, in general, a difference between group means that is greater than two standard errors is statistically significant. Thus one can “see” whether a difference is statistically significant based on a bar graph with error bars.

Figure 12.12 Sample APA-Style Bar Graph, With Error Bars Representing the Standard Errors, Based on Research by Ollendick and Colleagues

Line Graphs

Line graphs are used to present correlations between quantitative variables when the independent variable has, or is organized into, a relatively small number of distinct levels. Each point in a line graph represents the mean score on the dependent variable for participants at one level of the independent variable. Figure 12.13 “Sample APA-Style Line Graph Based on Research by Carlson and Conard” is an APA-style version of the results of Carlson and Conard. Notice that it includes error bars representing the standard error and conforms to all the stated guidelines.

Figure 12.13 Sample APA-Style Line Graph Based on Research by Carlson and Conard

In most cases, the information in a line graph could just as easily be presented in a bar graph. In Figure 12.13 “Sample APA-Style Line Graph Based on Research by Carlson and Conard” , for example, one could replace each point with a bar that reaches up to the same level and leave the error bars right where they are. This emphasizes the fundamental similarity of the two types of statistical relationship. Both are differences in the average score on one variable across levels of another. The convention followed by most researchers, however, is to use a bar graph when the variable plotted on the x- axis is categorical and a line graph when it is quantitative.

Scatterplots

Scatterplots are used to present relationships between quantitative variables when the variable on the x- axis (typically the independent variable) has a large number of levels. Each point in a scatterplot represents an individual rather than the mean for a group of individuals, and there are no lines connecting the points. The graph in Figure 12.14 “Sample APA-Style Scatterplot” is an APA-style version of Figure 12.8 “Statistical Relationship Between Several College Students’ Scores on the Rosenberg Self-Esteem Scale Given on Two Occasions a Week Apart” , which illustrates a few additional points. First, when the variables on the x- axis and y -axis are conceptually similar and measured on the same scale—as here, where they are measures of the same variable on two different occasions—this can be emphasized by making the axes the same length. Second, when two or more individuals fall at exactly the same point on the graph, one way this can be indicated is by offsetting the points slightly along the x- axis. Other ways are by displaying the number of individuals in parentheses next to the point or by making the point larger or darker in proportion to the number of individuals. Finally, the straight line that best fits the points in the scatterplot, which is called the regression line, can also be included.

Figure 12.14 Sample APA-Style Scatterplot

Expressing Descriptive Statistics in Tables

Like graphs, tables can be used to present large amounts of information clearly and efficiently. The same general principles apply to tables as apply to graphs. They should add important information to the presentation of your results, be as simple as possible, and be interpretable on their own. Again, we focus here on tables for an APA-style manuscript.

The most common use of tables is to present several means and standard deviations—usually for complex research designs with multiple independent and dependent variables. Figure 12.15 “Sample APA-Style Table Presenting Means and Standard Deviations” , for example, shows the results of a hypothetical study similar to the one by MacDonald and Martineau (2002) discussed in Chapter 5 “Psychological Measurement” . (The means in Figure 12.15 “Sample APA-Style Table Presenting Means and Standard Deviations” are the means reported by MacDonald and Martineau, but the standard errors are not). Recall that these researchers categorized participants as having low or high self-esteem, put them into a negative or positive mood, and measured their intentions to have unprotected sex. Although not mentioned in Chapter 5 “Psychological Measurement” , they also measured participants’ attitudes toward unprotected sex. Notice that the table includes horizontal lines spanning the entire table at the top and bottom, and just beneath the column headings. Furthermore, every column has a heading—including the leftmost column—and there are additional headings that span two or more columns that help to organize the information and present it more efficiently. Finally, notice that APA-style tables are numbered consecutively starting at 1 (Table 1, Table 2, and so on) and given a brief but clear and descriptive title.

Figure 12.15 Sample APA-Style Table Presenting Means and Standard Deviations

Another common use of tables is to present correlations—usually measured by Pearson’s r —among several variables. This is called a correlation matrix . Figure 12.16 “Sample APA-Style Table (Correlation Matrix) Based on Research by McCabe and Colleagues” is a correlation matrix based on a study by David McCabe and colleagues (McCabe, Roediger, McDaniel, Balota, & Hambrick, 2010). They were interested in the relationships between working memory and several other variables. We can see from the table that the correlation between working memory and executive function, for example, was an extremely strong .96, that the correlation between working memory and vocabulary was a medium .27, and that all the measures except vocabulary tend to decline with age. Notice here that only half the table is filled in because the other half would have identical values. For example, the Pearson’s r value in the upper right corner (working memory and age) would be the same as the one in the lower left corner (age and working memory). The correlation of a variable with itself is always 1.00, so these values are replaced by dashes to make the table easier to read.

Figure 12.16 Sample APA-Style Table (Correlation Matrix) Based on Research by McCabe and Colleagues

As with graphs, precise statistical results that appear in a table do not need to be repeated in the text. Instead, the writer can note major trends and alert the reader to details (e.g., specific correlations) that are of particular interest.

Key Takeaways

• In an APA-style article, simple results are most efficiently presented in the text, while more complex results are most efficiently presented in graphs or tables.
• APA style includes several rules for presenting numerical results in the text. These include using words only for numbers less than 10 that do not represent precise statistical results, and rounding results to two decimal places, using words (e.g., “mean”) in the text and symbols (e.g., “ M ”) in parentheses.
• APA style includes several rules for presenting results in graphs and tables. Graphs and tables should add information rather than repeating information, be as simple as possible, and be interpretable on their own with a descriptive caption (for graphs) or a descriptive title (for tables).
• Practice: In a classic study, men and women rated the importance of physical attractiveness in both a short-term mate and a long-term mate (Buss & Schmitt, 1993). The means and standard deviations are as follows. Men / Short Term: M = 5.67, SD = 2.34; Men / Long Term: M = 4.43, SD = 2.11; Women / Short Term: M = 5.67, SD = 2.48; Women / Long Term: M = 4.22, SD = 1.98. Present these results (a) in writing, (b) in a graph, and (c) in a table.

Buss, D. M., & Schmitt, D. P. (1993). Sexual strategies theory: A contextual evolutionary analysis of human mating. Psychological Review, 100 , 204–232.

MacDonald, T. K., & Martineau, A. M. (2002). Self-esteem, mood, and intentions to use condoms: When does low self-esteem lead to risky health behaviors? Journal of Experimental Social Psychology, 38 , 299–306.

McCabe, D. P., Roediger, H. L., McDaniel, M. A., Balota, D. A., & Hambrick, D. Z. (2010). The relationship between working memory capacity and executive functioning. Neuropsychology, 243 , 222–243.

• Research Methods in Psychology. Provided by : University of Minnesota Libraries Publishing. Located at : http://open.lib.umn.edu/psychologyresearchmethods/ . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike

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• Why Study Statistics?
• Descriptive & Inferential Statistics
• Fundamental Elements of Statistics
• Quantitative and Qualitative Data
• Measurement Data Levels
• Collecting Data
• Ethics in Statistics
• Describing Qualitative Data
• Describing Quantitative Data
• Stem-and-Leaf Plots
• Measures of Central Tendency
• Measures of Variability
• Describing Data using the Mean and Standard Deviation
• Measures of Position
• Counting Techniques
• Simple & Compound Events
• Independent and Dependent Events
• Mutually Exclusive and Non-Mutually Exclusive Events
• Permutations and Combinations
• Normal Distribution
• Central Limit Theorem
• Confidence Intervals
• Determining the Sample Size
• Hypothesis Testing
• Hypothesis Testing Process

How does the mean and standard deviation describe data?

The standard deviation is a measurement in reference to the mean that means:

• A large standard deviation indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean.
• When deciding whether sample measurements are suitable inferences for the population, the standard deviation of those measurements is of crucial importance.
• Standard deviations are often used as a measure of risk in finance associated with price-fluctuations of stocks, bonds, etc.

Chebyshev's rule  is an approximation of the percentage of data points captured between deviations of any data set. 

Example: A sample of size \(n=50\) has mean \(\bar{x}=28\) and standard deviation \(s=3\). Without knowing anything else about the sample, what can be said about the number of observations that lie in the interval \(922,34)\)? What can be said about the number of observations that lie outside the interval?

The interval \((22,34)\) is formed by adding and subtracting two standard deviations from the mean. By Chebyshev's Theorem, at least \(\frac{3}{4}\) of the data are within this interval. Since \(\frac{3}{4}\) of \(50\) is \(37.5\), this means that at least 37.5 observations are in the interval. But \(.5\) of a measurement does not make sense, so we conclude that at least 38 observations must lie inside the interval \((22,34)\).

If \(\frac{3}{4}\) of the observations are made inside the interval, than \(\frac{1}{4}\) of them are outside. We conclude that at most 12 \((50-38=12)\) observations lie outside the interval \((22,34)\).

There are more  accurate  ways of calculating the percentage or number of intervals inside standard deviations. Chebyshev's Theorem and the empirical rule we'll introduce next are just approximations.

If the histogram of a data set is approximately bell-shaped, we can approximate the percentage of data between standard deviations using the  empirical rule . 

Example: Heights of 18-yr-old males have a bell-shaped distribution with mean \(69.6\) inches and standard deviation \(1.4\) inches. About what proportion of all such mean are between 68.2 and 71 inches tall? And What interval centered on the mean should contain about 95% of all such mean?

Since the interval \((68.2,71.0)\) are one standard deviation from the mean, by the emprical rule, 68% of all 18-year old males have heights in this range.

95% by the empirical rule represents plus/minus two standard deviations from the mean.

\[\bar{x} \pm 2s = 69.6 \pm 2(1.4) = 66.8,\,72.4\]

Therefore, 95% of the mean are between 66.8 inches to 72.4 inches.

• Practice Questions - Empirical Rule

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• Next: Measures of Position >>
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Mean Deviation

How far, on average, all values are from the middle.

Calculating It

Find the mean of all values ... use it to work out distances ... then find the mean of those distances!

In three steps:

• 1. Find the mean of all values
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)
• 3. Then find the mean of those distances

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean :

Mean =   3 + 6 + 6 + 7 + 8 + 11 + 15 + 16 8   =   72 8   = 9

Step 2: Find the distance of each value from that mean:

Which looks like this:

(No minus signs!)

Step 3. Find the mean of those distances :

Mean Deviation =   6 + 3 + 3 + 2 + 1 + 2 + 6 + 7 8   =   30 8   = 3.75

So, the mean = 9 , and the mean deviation = 3.75

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For deviation just think distance

The formula is:

Mean Deviation = Σ|x − μ| N

• Σ is Sigma , which means to sum up
• || (the vertical bars) mean Absolute Value , basically to ignore minus signs
• x is each value (such as 3 or 16)
• μ is the mean (in our example μ = 9 )
• N is the number of values (in our example N = 8 )

Let's look at those in more detail:

Absolute Deviation

Each distance we calculate is called an Absolute Deviation , because it is the Absolute Value of the deviation (how far from the mean).

To show "Absolute Value" we put "|" marks either side like this:

For any value x :

Absolute Deviation = |x − μ|

From our example, the value 16 has:

Absolute Deviation = |x − μ| = |16 − 9| = |7| = 7

And now let's add them all up ...

The symbol for "Sum Up" is Σ (called Sigma Notation ), so we have:

Sum of Absolute Deviations = Σ|x − μ|

Divide by how many values N and we have:

Let's do our example again, using the proper symbols:

μ =   3 + 6 + 6 + 7 + 8 + 11 + 15 + 16 8   =   72 8   = 9

Step 2: Find the Absolute Deviations :

Step 3. Find the Mean Deviation :

Mean Deviation =   Σ|x − μ| N   =   30 8   = 3.75

Note: the mean deviation is sometimes called the Mean Absolute Deviation (MAD) because it is the mean of the absolute deviations.

What Does It "Mean" ?

Mean Deviation tells us how far, on average, all values are from the middle.

Here is an example (using the same data as on the Standard Deviation page):

Example: You and your friends have just measured the heights of your dogs (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

μ =   600 + 470 + 170 + 430 + 300 5   =   1970 5   = 394

Mean Deviation =   Σ|x − μ| N   =   636 5   = 127.2

So, on average, the dogs' heights are 127.2 mm from the mean .

(Compare that with the Standard Deviation of 147 mm )

A Useful Check

The deviations on one side of the mean should equal the deviations on the other side .

From our first example:

Example: 3, 6, 6, 7, 8, 11, 15, 16

The deviations are:

Example: Dogs

Deviations left of mean: 224 + 94 = 318

Deviations right of mean: 206 + 76 + 36 = 318

If they are not equal ... you may have made a msitake!

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Writers often need to discuss numbers and statistics in their manuscripts, and it can be a challenge to determine how to represent these in the most readable way. APA 7 contains detailed guidelines for how to write numbers and statistics, and the most common are listed below. These guidelines, however, are not exhaustive and writers may need to evaluate particular instances of numbers in their own writing to determine if the guideline applies or if an exception should be made for clarity.

Numbers 

Generally, you can spell out numbers below 10 in words (seven, three), and use numerals for anything 10 and higher (10, 42). 

• You should use Arabic numerals (1, 7) instead of Roman numerals (II, XI) unless the Roman numerals are part of established terminology in your field. 
• In numbers greater than 1,000, use commas to separate groups of three digits except in page numbers, binary code, serial numbers, temperatures, acoustic frequencies, and degrees of freedom. 
• Do not add apostrophes when writing a plural of a number (the 2000s, the 70s).

Use a numeral in these cases:

• a number 10 or higher anywhere in the paper
• a number right before a unit of measurement (3 m, 24 g)
• a number denoting: mathematical functions, fractions, decimals, percentages, ratios, percentiles (2:1 ratio, 5%)
• a number denoting: time, a date, an age, a point on a scale, an exact amount of money, or a numeral (the 3 key on your keyboard, 7 years old, a 5 on the test)
• a number indicating a place in a series or a part of a book/table, if the number is after a noun (i.e., Item 4, but words are used in cases like "the fourth item")

Spell the number out in words in these cases:

• a number from 0-9 anywhere in the paper, except the specific cases above
• a number that starts a sentence, heading, or title (though this should be avoided)
• a number that is a common fraction (one half, two thirds)
• a number that is part of a common phrase (Noble Eightfold Path)

When numbers are written next to each other in a sentence, one strategy to help readers parse the sentence is to combine words and numerals (3 two-year-old owls, four 3-step plans), but rewording to separate the numbers may be the best choice for clarity in some cases. Clarity for readers is always the most important consideration.

Ordinal Numbers 

Treat ordinal numbers (3rd, fourth) the same way as other numbers, using the guidelines above. You may use a superscript or not (1 st , 1 st ), but you should maintain the same usage throughout your paper.

Decimal Fractions 

In numbers less than 1, writers may include a leading 0 before the decimal point or not. This choice is based on the maximum possible amount of the statistic:

• If the statistic can be greater than 1, use a leading 0 (0.24 in)
• If the statistic cannot be greater than 1, do not use a leading 0 (p = .042)

APA's general principle for rounding decimals in experimental results is as follows, quoted here for accuracy: "Round as much as possible while considering prospective use and statistical precision" (7th edition manual, p. 180). Readers can more easily understand numbers with fewer decimal places reported, and generally APA recommends rounding to two decimal places (and rescaling data if necessary to achieve this).

Some more specific guidelines for particular values are listed below.

One decimal place:

• standard deviations

Two decimal places:

• correlations
• proportions
• inferential statistics
• exact p values (can be reported to two or three places; when p is less than .001, write p < .001)

Statistical Copy 

These rules cover presentation of data, not accuracy of data or the best way to conduct analysis.

You can represent data in the text, in a table, or in a figure. A rule of thumb is:

• <3 numbers → try a sentence
• 4-20 numbers → try a table
• >20 numbers → try a figure

Clarity is always paramount.

When discussing statistics in common use, you do not need to provide a reference or formula.

If the statistic or expression is new, rare, integral to the paper, or used in an unconventional way, provide a reference or formula.

The purpose of reporting statistics is usually to help readers confirm your findings and analyses; as such, the degree of specificity in reporting results should follow in line with that purpose.

When your data are multilevel, you should include summary statistics for each level, depending on the kind of analysis performed. When your data are reported in a table or figure, you do not need to repeat each number in the text, but you should mention the table or figure in the text when discussing the statistics and emphasize in-text key data points that help interpret your findings. 

Use words like "respectively" or "in order" to clarify each statistic mentioned in text and their referent.

For instance:

Confidence intervals should be reported: 90% CI [ LL, UL ], with LL as the lower limit and UL as the upper limit of the interval. You do not need to repeat confidence intervals in the same paragraph or in a series when the meaning is clear and the confidence interval has not changed. When CIs follow the report of a point estimate, you do not need to repeat the unit of measurement.

Statistics uses a great deal of symbols and abbreviations (when a term can be both, the abbreviation refers to the concept and the symbol indicates a numeric value).

You do not need to define these when they represent a statistic or when they are composed of Greek letters. You do need to define any other abbreviation (such as ANOVA, CFA, SEM) in your paper. If the analysis you are performing uses multiple notation styles for symbols and abbreviations, only use one consistently throughout your paper.

Some other statistical symbol guidelines include:

• use words rather than symbols in narrative text; when you report a stats term with other mathematical symbols like = or +, use the symbol
• population parameters use Greek letters while estimators use Latin letters in italics (usually)
• uppercase, italicized N indicates the total membership of a sample; lowercase, italicized n indicates the membership of a subgroup of a sample such as a treatment group or control group
• % and currency symbols like $ should only be used with numerals (15%, $25) or in table headings and figure labels to save space
• use standard type (no italics or bold) for Greek letters, subscript and superscript identifiers, and abbreviations that are not variables such as log
• use bold type for vector and matrix symbols
• use italics for all other statistical symbols

Mathematical Copy 

For ease of reading, use spaces between elements in a mathematical expression ( a + b = c ), except in the case of a minus sign indicating a negative number which uses a space before the minus but not between the minus and the numeral.

Use subscripts first and then superscripts, except in the case of key symbols like the superscript for prime.

All equations should be punctuated to fit in the syntax of the sentence, even if they are presented on their own line.

Short, simple equations can be written in a regular line of text, with a slash (/) for fractions. Parentheses, square brackets, and braces should be used (in that order, from innermost to outermost) to indicate order of operations. Equations that do not fit vertically in the line of text should be shown on their own line. 

All displayed equations (equations on their own line) should be numbered, similarly to tables and figures, so that they can be referred to later (and simple equations may be displayed rather than written in a line of text if they will need to be referred to later by number).

In text, equations should be referred to by name (Equation 1 or the first equation are both acceptable). The equation number does not need a special label, and instead should be displayed in parentheses toward the right margin of the page:

  If a symbol in your equation cannot be entered with your word processor, use an image; otherwise, type all equations exactly as you would like them to appear in the publication.

• Range and Mean Deviation

The field of statistics has practical applications in almost all fields of life. In finance and investing and manufacturing and various other fields. Whether you want to launch a rocket or calculate a students performance we take the help of statistics . Now let us learn the concepts of range and mean deviation.

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For a given series of data, statistics aims at analysis and drawing conclusions . The various measures of central tendency – mean, median and mode represent the values in a series. However, we can further implement this analytical claim of statistics, by measuring the scattering and dispersion of data around these measures of central tendency.

For example, the range, in a series of men distributed by their ages, gives us an idea of how much the ages can vary from the central value. So without further ado, let us jump right into the concepts of range and mean deviation.

Browse more Topics under Statistics

• Bar Graphs and Histogram
• Cumulative Frequency Curve
• Frequency Distribution
• Frequency Polygon
• Range and Mean Deviation for Grouped Data
• Range and Mean Deviation for Ungrouped Data
• Variance and Standard Deviation

You can download Cheat Sheet of Statistics by clicking on the download button below

The range can be simply understood as the value that tells us about the scattering of data. This gives us an idea of how much the data can vary. Consequently, it is related to the maximum and minimum values in a distribution . The range is the difference between the maximum and the minimum value in a distribution. Notably, it only gives us the idea of the spread of data . It does not tell us about the dispersion of values from a measure of central tendency.

Range = Maximum value – Minimum value

Mean Deviation

To understand the dispersion of data from a measure of central tendency, we can use mean deviation. It comes as an improvement over the range. It basically measures the deviations from a value. This value is generally mean or median. Hence although mean deviation about mode can be calculated, mean deviation about mean and median are frequently used.

Note that the deviation of an observation from a value a is  d= x-a.  To find out mean deviation we need to take the mean of these deviations. However, when this value of a is taken as mean, the deviations are both negative and positive since it is the central value.

This further means that when we sum up these deviations to find out their average , the sum essentially vanishes. Thus to resolve this problem we use absolute values or the magnitude of deviation. The basic formula for finding out mean deviation is :

Mean deviation= Sum of absolute values of deviations from ‘a’ ÷ The number of observations

Solved Example for You

Q: The sum of squares of deviation of variates from their A.M. is always

• Cannot be said

Sol: The correct option is “B”. It is a fundamental concept that the sum of squares of deviation of any variate from their arithmetic mean is always minimum.

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• Knowledge Base
• Central Tendency | Understanding the Mean, Median & Mode

Central Tendency | Understanding the Mean, Median & Mode

Published on July 30, 2020 by Pritha Bhandari . Revised on June 21, 2023.

Measures of central tendency help you find the middle, or the average, of a dataset. The 3 most common measures of central tendency are the mode, median, and mean.

• Mode : the most frequent value.
• Median : the middle number in an ordered dataset.
• Mean : the sum of all values divided by the total number of values.

In addition to central tendency, the variability and distribution of your dataset is important to understand when performing descriptive statistics .

Table of contents

Distributions and central tendency, when should you use the mean, median or mode, other interesting articles, frequently asked questions about central tendency.

A dataset is a distribution of n number of scores or values.

Normal distribution

In a normal distribution , data is symmetrically distributed with no skew . Most values cluster around a central region, with values tapering off as they go further away from the center. The mean, mode and median are exactly the same in a normal distribution.

A histogram of your data shows the frequency of responses for each possible number of books. From looking at the chart, you see that there is a normal distribution.

Skewed distributions

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other. The direction of this tail tells you the side of the skew

In a positively skewed distribution, there’s a cluster of lower scores and a spread out tail on the right. In a negatively skewed distribution, there’s a cluster of higher scores and a spread out tail on the left.

• Positively skewed distribution
• Negatively skewed distribution

In this histogram, your distribution is skewed to the right, and the central tendency of your dataset is on the lower end of possible scores.

In a positively skewed distribution, mode < median < mean.

In a negatively skewed distribution, mean < median < mode.

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The mode is the most frequently occurring value in the dataset. It’s possible to have no mode, one mode, or more than one mode.

To find the mode, sort your dataset numerically or categorically and select the response that occurs most frequently.

To find the mode, sort your data by category and find which response was chosen most frequently.

To make it easier, you can create a frequency table to count up the values for each category.

Mode: Liberal

The mode is easily seen in a bar graph because it is the value with the highest bar.

When to use the mode

The mode is most applicable to data from a nominal level of measurement. Nominal data is classified into mutually exclusive categories, so the mode tells you the most popular category.

For continuous variables or ratio levels of measurement, the mode may not be a helpful measure of central tendency. That’s because there are many more possible values than there are in a nominal or ordinal level of measurement. It’s unlikely for a value to repeat in a ratio level of measurement .

The median of a dataset is the value that’s exactly in the middle when it is ordered from low to high.

To find the median, you first order all values from low to high. Then, you find the value in the middle of the ordered dataset—in this case, the value in the 4th position.

Median: Medium

In larger datasets, it’s easier to use simple formulas to figure out the position of the middle value in the distribution. You use different methods to find the median of a dataset depending on whether the total number of values is even or odd.

Median of an odd-numbered dataset

That means the median is the 3rd value in your ordered dataset.

Median: 345 milliseconds

Median of an even-numbered dataset

That means the middle values are the 3rd value, which is 345 , and the 4th value, which is 357 .

To get the median, take the mean of the 2 middle values by adding them together and dividing by 2.

Median: 351 milliseconds

The arithmetic mean of a dataset (which is different from the geometric mean ) is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency because all values are used in the calculation.

First you add up the sum of all values:

Then you calculate the mean using the formula

There are 5 values in the dataset, so n = 5.

Mean (x̄ ) : 335 milliseconds

Outlier effect on the mean

Outliers can significantly increase or decrease the mean when they are included in the calculation. Since all values are used to calculate the mean, it can be affected by extreme outliers. An outlier is a value that differs significantly from the others in a dataset.

Mean: 444 milliseconds

Population versus sample mean

A dataset contains values from a sample or a population . A population is the entire group that you are interested in researching, while a sample is only a subset of that population.

While data from a sample can help you make estimates about a population, only full population data can give you the complete picture.

In statistics, the notation of a sample mean and a population mean and their formulas are different. But the procedures for calculating the population and sample means are the same.

• x̄:  sample mean

• n: number of values in the sample dataset

• μ: population mean

• N: number of values in the population dataset

The 3 main measures of central tendency are best used in combination with each other because they have complementary strengths and limitations. But sometimes only 1 or 2 of them are applicable to your dataset, depending on the level of measurement of the variable.

• The mode can be used for any level of measurement, but it’s most meaningful for nominal and ordinal levels.
• The median can only be used on data that can be ordered – that is, from ordinal, interval and ratio levels of measurement.
• The mean can only be used on interval and ratio levels of measurement because it requires equal spacing between adjacent values or scores in the scale.

To decide which measures of central tendency to use, you should also consider the distribution of your dataset.

For normally distributed data, all three measures of central tendency will give you the same answer so they can all be used.

In skewed distributions, the median is the best measure because it is unaffected by extreme outliers or non-symmetric distributions of scores. The mean and mode can vary in skewed distributions.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Statistical power
• Pearson correlation
• Degrees of freedom
• Statistical significance

Methodology

• Cluster sampling
• Stratified sampling
• Focus group
• Systematic review
• Ethnography
• Double-Barreled Question

Research bias

• Implicit bias
• Publication bias
• Cognitive bias
• Placebo effect
• Pygmalion effect
• Hindsight bias
• Overconfidence bias

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

• The mode is the most frequent value.
• The median is the middle number in an ordered data set.
• The mean is the sum of all values divided by the total number of values.

The measures of central tendency you can use depends on the level of measurement of your data.

• For a nominal level, you can only use the mode to find the most frequent value.
• For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.
• For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

Cite this Scribbr article

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Bhandari, P. (2023, June 21). Central Tendency | Understanding the Mean, Median & Mode. Scribbr. Retrieved August 19, 2024, from https://www.scribbr.com/statistics/central-tendency/

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How to Calculate Mean Deviation About Mean (for Ungrouped Data)

Last Updated: December 15, 2023 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 533,787 times.

In working with data, there are several different ways to measure how closely grouped your data values are. The most common is the mean. Most people learn early in school to calculate the mean by finding the sum of a group of data values and then dividing by the number of values in the set. A more advanced calculation is the mean deviation about the mean. This calculation tells you how close to the mean your values are. Finding this consists of finding the mean for a data set, finding the difference of each data point from that mean, and then taking the mean of those differences.

Calculating the Mean

• For this example, use the assigned data set of 6, 7, 10, 12, 13, 4, 8 and 12. This set is small enough to count by hand to find that there are eight numbers in the set.

Finding the Mean Deviation

• Fill the first column with the data points for your calculation.

• To check the validity of your calculations, the sum of the values in this deviation column should be 0. If you add them up and get something other than 0, then either your mean is incorrect or you made an error in calculating one or more of the deviations. Go back and check your work.

• Absolute value is a mathematical tool used to measure distance or size, regardless of direction.
• To find absolute value, just drop the negative sign from each number in the second column. Thus, fill the third column with the absolute values as follows:

• For example, with this data set, you can say that the mean is 9 and the average distance from that mean is 2.75. Note that some numbers are closer than 2.75 and some are farther. But that is the average distance.

Community Q&A

• Keep practicing and you will be able to do it quickly. Thanks Helpful 7 Not Helpful 3

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• ↑ http://www.mathsisfun.com/definitions/mean.html
• ↑ https://www.mathsisfun.com/mean.html
• ↑ http://www.mathsisfun.com/data/mean-deviation.html
• ↑ https://www.mathsisfun.com/data/mean-deviation.html
• ↑ https://sciencing.com/absolute-deviation-average-absolute-deviation-4918826.html
• ↑ https://www.cuemath.com/mean-deviation-formula/

About This Article

To calculate mean deviation about mean for ungrouped data, start by finding the mean of your data set by adding all of the data points together and then dividing by the total number of points. Once you have the mean, calculate the deviation of each data point by subtracting the mean from each point. Then, drop the negative sign from any deviations that have them. Finally, calculate the mean of the deviations by adding them together and dividing by the total number of deviations. To see an example problem, keep reading! Did this summary help you? Yes No

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Mean Deviation About Median

In Statistics, Mean deviation is one of the measures of dispersion. This can be obtained from any one of the measures of central tendency. Though, mean deviation from mean and median are the most commonly used statistical considerations. In this article, one can learn what is mean deviation about median, how to find mean deviation about median in case of both grouped and ungrouped data along with examples.

Mean Deviation About The Median

The mean deviation about a median value ‘M’ is the mean of the absolute values of the deviations of the observations from ‘M’. The mean deviation from ‘M’ is denoted as M.D. (M).

Mean Deviation About Median Formula

The mean deviation from the median formula is given by:

Here, “M” represents the median of the given data. However, we can find the mean deviation from the median for ungrouped and grouped data as explained below.

How to Calculate Mean deviation from median

The three main steps involved in finding the mean deviation about median either for ungrouped or grouped data are given below:

Step 1: Find the median value for the given data values.

Step 2: Now, subtract the median value from each of the data values given and consider the absolute deviations here.

Step 3: Finally, calculate the mean of deviations obtained in the previous step.

Mean Deviation About Median For Ungrouped Data

Let x 1 , x 2 , x 3 ,…, x n be n observations. For this data, the formula to find the mean deviation from the median is given as:

M = Median of the given data

Click here to learn how to find the median of numbers .

Let’s have a look at the example given below for better understanding.

Find the mean deviation about the median for the data: 6, 7, 10, 12, 13, 4, 8, 12

Given data:

6, 7, 10, 12, 13, 4, 8, 12, 16

Finding the median:

Ascending order of the given data is: 4, 6, 7, 8, 10, 12, 12, 13, 16

Number of data values = 9

Median = (n + 1)/2 th observation

= (9 + 1)/2

= 5th observation

Thus, median = 10

The absolute values of the respective deviations from the median, i.e., |xi − M| are:

|4 – 10|, |6 – 10|, |7 – 10|, |8 – 10|, |10 – 10|, |12 – 10|, |12 – 10|, |13 – 10|, |16 – 10|

= 6, 4, 3, 2, 0, 2, 2, 3, 6

= (6 + 4 + 3 + 2 + 0 + 2 + 2 + 3 + 6)/9

Therefore, the mean deviation about the median for the given data is 3.11.

Mean Deviation About Median For Grouped Data

For grouped data, there exist two scenarios where we can find the median deviation from the median. They are:

(i) Mean Deviation About Median for Discrete frequency distribution

To find the mean deviation about the median, we need to calculate the median of the given discrete frequency distribution. As we already know how to find the median of data in case of discrete distribution, we can continue finding the mean deviation using the formula given below:

x i = Data values

f i = Frequency

N = Sum of the frequencies

Go through the example given below to understand the process clearly.

Example 2: Find mean deviation about the median for the following distribution.

The first step in calculating the median for discrete distribution is to arrange the data in ascending order.

As the given data is already in ascending order,we can find the median as:

N = 40 (even)

N/2 = 40/2 = 20

Median will be the average of 20th and 21st observations.

However, these observations lie in the cumulative frequency (cf) 23, for which the corresponding observation is 6.

Median = (6 + 6)/2 = 6

Let’s prepare another table for getting the absolute deviations.

We know that,

= (8 + 9 + 0 + 8 + 32 + 30)/40

Therefore, the mean deviation about the median for the given distribution is 2.175.

(ii) Mean Deviation About Median for Continuous frequency distribution

The mean deviation from the median of continuous frequency distribution can be calculated using the following formula.

x i = Mid-point of the class interval

For continuous frequency distribution, the formula of median is given as:

l = Lower limit of median class

N = Number of observations (sum of the frequencies)

C = Cumulative frequency of class preceding the median class

f = Frequency of median class

h = Class size (assuming class size to be equal)

Given below is the solved example to learn how to find the mean deviation about median for continuous distribution of data.

Example 3: Calculate the mean deviation about the median for the following distribution.

Let us calculate the median and mean deviation for the given data.

N = ∑f i = 40

The cumulative frequency greater than or equal to N/2 is 27, for which the corresponding class interval is 40 – 50.

Median class = 40 – 50

Median = l + [(N/2 – C)/f] × h

= 40 + [(20 – 13)/14] × 10

= 40 + (7/14) × 10

Therefore, medan (M) = 45

From the above table,

∑f i |x i – M| = 400

Hence, the required mean deviation = 400/40 = 10

Video Lesson on Median of Data

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Measures of Dispersion | Types, Formula and Examples

Measures of Dispersion are used to represent the scattering of data. These are the numbers that show the various aspects of the data spread across various parameters.

Let’s learn about the measure of dispersion in statistics , its types, formulas, and examples in detail.

Dispersion in Statistics

Dispersion in statistics is a way to describe how spread out or scattered the data is around an average value. It helps to understand if the data points are close together or far apart.

Dispersion shows the variability or consistency in a set of data. There are different measures of dispersion like range, variance, and standard deviation.

Measure of Dispersion in Statistics

Measures of Dispersion measure the scattering of the data. It tells us how the values are distributed in the data set. In statistics, we define the measure of dispersion as various parameters that are used to define the various attributes of the data.

These measures of dispersion capture variation between different values of the data.

Types of Measures of Dispersion

Measures of dispersion can be classified into the following two types :  

Absolute Measure of Dispersion

Relative measure of dispersion.

These measures of dispersion can be further divided into various categories. They have various parameters and these parameters have the same unit.

Let’s learn about them in detail.

The measures of dispersion that are measured and expressed in the units of data themselves are called Absolute Measure of Dispersion. For example – Meters, Dollars, Kg, etc.

Some absolute measures of dispersion are: 

Range: It is defined as the difference between the largest and the smallest value in the distribution.

Mean Deviation: It is the arithmetic mean of the difference between the values and their mean.

Standard Deviation: It is the square root of the arithmetic average of the square of the deviations measured from the mean.

Variance: It is defined as the average of the square deviation from the mean of the given data set.

Quartile Deviation : It is defined as half of the difference between the third quartile and the first quartile in a given data set.

Interquartile Range: The difference between upper(Q 3 ) and lower(Q 1 ) quartile is called Interterquartile Range. Its formula is given as Q 3 – Q 1.

Read More :

• Mean deviation
• Standard Deviation
• Quartile Deviation

We use relative measures of dispersion to measure the two quantities that have different units to get a better idea about the scattering of the data.

Here are some of the relative measures of dispersion:

Coefficient of Range: It is defined as the ratio of the difference between the highest and lowest value in a data set to the sum of the highest and lowest value.

Coefficient of Variation: It is defined as the ratio of the standard deviation to the mean of the data set. We use percentages to express the coefficient of variation.

Coefficient of Mean Deviation: It is defined as the ratio of the mean deviation to the value of the central point of the data set.

Coefficient of Quartile Deviation: It is defined as the ratio of the difference between the third quartile and the first quartile to the sum of the third and first quartiles.

• Coefficient of Mean Deviation
• Coefficient of Variation
• Coefficient of Range

Range of Data Set

The range is the difference between the largest and the smallest values in the distribution.

Thus, it can be written as

R = L – S

L is the largest value in the Distribution

S is the smallest value in the Distribution

• A higher value of range implies higher variation in the data set.
• One drawback of this measure is that it only takes into account the maximum and the minimum value. They might not always be the proper indicator of how the values of the distribution are scattered. 

Example: Find the range of the data set 10, 20, 15, 0, 100.

Smallest Value in the data = 0 Largest Value in the data = 100  Thus, the range of the data set is, R = 100 – 0 R = 100

Note: Range cannot be calculated for the open-ended frequency distributions. Open-ended frequency distributions are those distributions in which either the lower limit of the lowest class or the higher limit of the highest class is not defined. 

Range for Ungrouped Data

To find the range for the ungrouped data set, first we have to find the smallest and the largest value of the data set by observing. The difference between them gives the range of ungrouped data.

We can understyand this with the help of following example:

Example: Find out the range for the following observations, 20, 24, 31, 17, 45, 39, 51, 61.

Largest Value = 61 Smallest Value = 17 Thus, the range of the data set is Range = 61 – 17 = 44

Range for Grouped Data

The range of the grouped data set is found by studying the following example,

Example: Find out the range for the following frequency distribution table for the marks scored by class 10 students. 

For Largest Value: Taking the higher limit of Highest Class = 40  For Smallest Value: Taking the lower limit of Lowest Class = 0 Range = 40 – 0  Thus, the range of the given data set is, Range = 40 

Mean Deviation

Mean deviation measures the deviation of the observations from the mean of the distribution.

Since the average is the central value of the data, some deviations might be positive and some might be negative. If they are added like that, their sum will not reveal much as they tend to cancel each other’s effect.

For example :

Let us consider this set of data : -5, 10, 25

Mean = (-5 + 10 + 25)/3 = 10

Now a deviation from the mean for different values is,

• (-5 -10) = -15
• (10 – 10) = 0
• (25 – 10) = 15

Now adding the deviations, shows that there is zero deviation from the mean which is incorrect. Thus, to counter this problem only the absolute values of the difference are taken while calculating the mean deviation.

Mean Deviation Formula :

Mean Deviation for Ungrouped Data

For calculating the mean deviation for ungrouped data, the following steps must be followed: 

Step 1: Calculate the arithmetic mean for all the values of the dataset.

Step 2: Calculate the difference between each value of the dataset and the mean. Only absolute values of the differences will be considered. |d|

Step 3: Calculate the arithmetic mean of these deviations using the formula,

This can be explained using the example.

Example: Calculate the mean deviation for the given ungrouped data, 2, 4, 6, 8, 10

Mean(μ) = (2+4+6+8+10)/(5) μ = 6 M. D =  ⇒ M.D =  ⇒ M.D = (4+2+0+2+4)/(5) ⇒ M.D = 12/5 = 2.4

Read More On :

• Arithmetic Mean

Measures of Dispersion Formula

Measures of Dispersion Formulas are used to tell us about the various parameters of the data. Various formulas related to the measures of dispersion are discussed in the table below.

Coefficient of Dispersion

Coefficients of dispersion are calculated when two series are compared, which have great differences in their average. We also use co-efficient of dispersion for comparing two series that have different measurements. It is denoted using the letters C.D.

Measures of Dispersion and Central Tendency

Both Measures of Dispersion and Central Tendency are numbers that are used to describe various parameters of the data. Let’s see the differences between Measures of Dispersion and Central Tendency.

• Central Tendency
• Quartile Formula
• Statistics Formulas
• Difference Between Mean, Median and Mode
• Probability and Statistics
• Variance and Standard Deviation

Examples on Measures of Dispersion

Let’s solve some questions on the Measures of Dispersion.

Examples 1: Find out the range for the following observations. {20, 42, 13, 71, 54, 93, 15, 16}

Given, Largest Value of Observation = 71 Smallest Value of Observation = 13 Thus, the range of the data set is, Range = 71 – 13 Range = 58

Example 2: Find out the range for the following frequency distribution table for the marks scored by class 10 students. 

Given, Largest Value: Take the Higher Limit of the Highest Class = 40  Smallest Value: Take the Lower Limit of the Lowest Class = 10 Range = 40 – 10 Range = 30 Thus, the range of the data set is 30.

Example 3: Calculate the mean deviation for the given ungrouped data {-5, -4, 0, 4, 5}

Mean(μ) = {(-5)+(-4)+(0)+(4)+(5)}/5 μ = 0/5 = 0 M. D =  ⇒ M.D =  ⇒ M.D = (5+4+0+4+5)/5 ⇒ M.D = 18/5 ⇒ M.D = 3.6

Measures of Dispersion- FAQs

What is measure of dispersion in statistics.

Measure of Dispersion is the positive real numbers that are used to define the variability of the data set about any central point.

What are Types of Measures of Dispersion in Statistics?

Measures of Dispersion are classified into two types : Absolute Measures of Dispersion Relative Measures of Dispersion

What is Absolute Measure of Dispersion?

Absolute Measures of Dispersion are the statistical tools that provide the actual spread of data, like range and standard deviation. They have the same units as the data.

What is Relative Measure of Dispersion?

Relative Measure of Dispersions show the spread of data relative to its central value, without unit dependency. They are statistical comparisons, expressed as ratios or percentages, like the coefficient of variation.

What is the Difference between Absolute and Relative Measure of Dispersion?

Absolute measures of dispersion provide the actual spread of data, like range, variance, standard deviation. They are expressed in the same units as the data. Relative measures of dispersion, on the other hand, compare the spread relative to the central value, usually as a ratio or percentage. They are unitless, allowing for comparison between different datasets.

How to Calculate Dispersion in Statistics?

Dispersion is calculated by using various formulas for mean, standard deviation, variance, etc.

What are Examples of Dispersion in Statistics?

Examples of dispersion in statistics include: Range, Variance, Standard Deviation, Interquartile Range (IQR), Coefficient of Variation, etc.

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Donald J. Trump intends to bring independent regulatory agencies under direct presidential control. Credit... Doug Mills/The New York Times

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