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Statistics Calculator: Mean Absolute Deviation (MAD)

Use this calculator to compute the mean absolute deviation from a data set.

Mean Absolute Deviation Calculator

Instructions.

This calculator computes the mean absolute deviation from a data set:

You do not need to specify whether the data is for an entire population or from a sample. Just type or paste all observed values in the box above. Values must be numeric and may be separated by commas, spaces or new-line. Press the "Submit Data" button to perform the computation. To clear the calculator, press "Reset".

What is the mean absolute deviation

The mean deviation is a measure of dispersion , A measure of by how much the values in the data set are likely to differ from their mean. The absolute value is used to avoid deviations with opposite signs cancelling each other out.

Mean absolute deviation formula

Mean Absolute Deviation Calculator

What is mean absolute deviation mean absolute deviation calculator, how to find mean absolute deviation (mad) mean absolute deviation formula, how to find mad - an example.

The mean absolute deviation calculator is a tool that can help you quickly find the mean absolute deviation around the mean, median, or any other number. If you want to know what is mean absolute deviation - we break down the name in the article below. We also explain how to find it using a mean absolute deviation formula, or by following a few simple steps.

Now, input up to 50 numbers and get the mean absolute deviation for your dataset!

The mean absolute deviation shows us how spread out the numbers are in a data set. But how? And what does deviation even mean?

• Deviation is the difference from what is usual .
• An absolute number takes the positive value of a number, without regards to its sign.
• Mean is an average of a set of numbers.

So, what is the mean absolute deviation? It's the average of every value's distance from a certain central point . This point can be a mean, median, mode , or any other statistically significant number.

To calculate it, use our mean absolute deviation calculator or do it on your own using the How to find mean absolute deviation? section below. How to use the MAD calculator? Select the central point. If you want it to be mean or median of your data set - the calculator will calculate it for you. If not, choose the option other and input an appropriate value. All that's left for you to do is input your data set and get your result!

If you are wondering how to find MAD, our calculator uses the mean absolute deviation formula:

M A D = 1 n ∑ i = 1 n ∣ x i − m ∣ \small MAD = \frac{1}{n}\sum_{i=1}^n|x_i-m| M A D = n 1 ​ ∑ i = 1 n ​ ∣ x i ​ − m ∣ ,

• n is the amount of numbers in the set;
• x i is the i th number of the set; and
• m is a certain central point (mean, median, mode, etc)

So, to find mean absolute deviation, follow these steps:

Determine the central point: m \footnotesize m m .

Find the difference from the central point for each point: x i − m \footnotesize x_i - m x i ​ − m .

Calculate the absolute value of each difference: ∣ x i − m ∣ \footnotesize |x_i - m| ∣ x i ​ − m ∣ .

Find the mean of the absolute deviations - sum up all the absolute differences and divide them by the amount of data points: 1 n ∑ ∣ x i − m ∣ \footnotesize\frac{1}{n}\sum|x_i-m| n 1 ​ ∑ ∣ x i ​ − m ∣ .

Now, let's go to an example to see how to find the mean absolute deviation by using these steps.

Let's calculate the mean absolute deviation for 6 friends' lost socks in the last year: 3, 17, 9, 7, 13, 11.

• Let's say you want to calculate the MAD around the mean. In this case, the mean is the amount of socks lost, on average, by one person:

m = (3 + 17 + 9 + 7 + 13 + 11) / 6 = 60 / 6 = 10

• Calculate the deviation from the mean:
• 3 - 10 = -7
• 17 - 10 = 7
• 9 - 10 = -1
• 7 - 10 - -3
• 13 - 10 = 3
• 11 - 10 = 1
• Find the absolute value of each deviation:
• Find the mean of absolute values of each deviation:

MAD = (7 + 7 + 1 + 3 + 3 + 1) / 6 = 3.67

It means that, on average, the amount of lost socks differs from the mean by 3.67 "socks". What does it mean? It means that, in theory, each of 6 friends lost ten socks in the last year on average, while in reality this isn't the case - they lost anywhere between 6.33 and 13.67 socks (10±3.67).

Even if you now know how to find MAD by hand, you don't need to repeat these tedious calculation anymore. Just use our mean absolute deviation calculator!

• MAD can indicate mean absolute deviation or median absolute deviation . These are not the same. In the median absolute deviation , you calculate the median of the distances from a central point.
• The distribution of data points can be also portrayed with standard deviation. Although similar in concept, standard deviation gives us the opportunity to further analyze a dataset with variance , estimate sample size , and so on.

Now you know what is mean absolute deviation and how to calculate it. Before you go, check out our other descriptive statistics calculators like this percentile calculator !

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Mean Absolute Deviation Worksheets | Find the Mean and MAD

Walk through this compilation of printable mean absolute deviation worksheets, hand-picked for students of grade 6 and grade 7, to bolster skills in finding the average absolute deviation of data sets up to 6 and up to 10 offering three levels each. Level 1 features whole numbers up to 99, Level 2 involves 2-digit and 3-digit whole numbers while Level 3 comprises decimals. Learn to compare data sets and apply skills acquired in solving word problems. A thorough knowledge of calculating the mean is a prerequisite in solving the pdf worksheets presented here. Free worksheets are also included.

Mean Absolute Deviation | Up to 6 Data Sets | Level 1

Determine the mean and deviation (the difference between each data value and the mean). Find the absolute or positive values and the averages of these differences to compute the MAD for each data set.

• Download the set

Mean Absolute Deviation | Up to 6 Data Sets | Level 2

Level up with data values involving 2-digit and 3-digit whole numbers. Compute the mean, the distance and the average of the distances. Complete four tables and solve a word problem in each printable worksheet.

Mean Absolute Deviation | Up to 6 Data Sets | Level 3 - Decimals

Direct 6th grade students to find the average of the positive distances of each data value from the central point or mean for the sets of decimal data values presented here using the mean absolute deviation formula.

Mean Absolute Deviation | Up to 10 Data Sets | Level 1

Intensify your practice in finding the mean absolute deviation of data sets up to 10, with these middle school worksheets. Find the mean and the average of the distances using the table templates provided.

Mean Absolute Deviation | Up to 10 Data Sets | Level 2

Plug in the mean, absolute deviation and the number of data values in the mean absolute deviation formula to solve the word problem and the sets of data values offered in each mean absolute deviation worksheet.

Mean Absolute Deviation | Up to 10 Data Sets | Level 3 - Decimals

Packed in each worksheet are 6 sets of decimal data values. Determine the mean absolute deviation of each set and round the answers to two decimal places. Review concept by solving a word problem.

Comparing the Mean Absolute Deviations for Two Data Sets

Compare two distributions of data by calculating their means and MADs in these 7th grade worksheet pdfs. A larger MAD indicates a widely spread out data while a smaller MAD implies clustered data.

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Mean Absolute Deviation

To understand Mean Absolute Deviation,  let us split both the words and try to figure out their meaning. ‘Mean’ refers to the average of the observations and deviation implies departure or variation from a preset standard.

When put together, we can define mean deviation as the mean distance of each observation from the mean of the data.

Mean Absolute Deviation Formula

Ratio of sum of all absolute values of deviation from central measure to the total number of observations.

M.A. D = (Σ Absolute Values of Deviation from Central Measure) / (Total Number of Observations)

Calculate Mean Absolute Deviation

Steps to find the mean deviation from mean:

(i)Find the mean of the given observations.

(ii)Calculate the difference between each observation and the calculated mean

(iii)Evaluate the mean of the differences obtained in the second step.

This gives you the mean deviation from mean.

Suppose that the deviation from a central value a is given as (x-a), where x is any observation of the set of data. To find out the mean deviation, we need to find the average of all the deviations from a in the given data set. Since the measure of central tendency lies between the maximum and minimum values of the data set,we can see that some deviations would be positive and rest would be negative.The sum of such deviations would give a zero. Let us see an example to make this point clearer to you.

Consider the following data set

The mean of the given data is given as:

\(\begin{array}{l}\overline{x}\end{array} \) = \(\begin{array}{l}\frac{\sum\limits_{i=1}{n}x_i}{n}\end{array} \)

\(\begin{array}{l} \Rightarrow \overline{x} \end{array} \) = \(\begin{array}{l} \frac{85 + 75 + 80}{3} \end{array} \) = \(\begin{array}{l} 80 \end{array} \)

Now, if we calculate the deviation from mean for the given values, we have:

From the definition , we have

\(\begin{array}{l}Mean~ Deviation\end{array} \) = \(\begin{array}{l} \frac{Sum~ of~ all~ the~ deviations~ from~ mean}{Total ~number~ of ~observations}\end{array} \)

\(\begin{array}{l}\Rightarrow Mean~ Deviation\end{array} \) = \(\begin{array}{l}{5+(-5)+0} \end{array} \) = \(\begin{array}{l}0\end{array} \)

This does not give us any idea about measure of variability of the data which is the actual purpose of finding the mean deviation. So, we find the absolute value of deviation from the mean.

In the above example the mean absolute deviation can be calculated as:

\(\begin{array}{l}Mean ~Absolute~ Deviation~(M.A.D)\end{array} \) = \(\begin{array}{l}\frac{5+|-5|+0}{3}\end{array} \) = \(\begin{array}{l}\frac{10}{3}\end{array} \) = \(\begin{array}{l}3.333\end{array} \)

This gives us an idea about the deviation of the observations from the measure of central tendency.

Thus we can conclude that,

\(\begin{array}{l}M.A.D\end{array} \) = \(\begin{array}{l}\frac{∑Absolute ~values ~of ~Deviation ~from ~cental~ measure}{Total~ Number~ of ~observations}\end{array} \)

Although to calculate mean absolute deviation, any measure of central tendency can be used but generally mean and median are the most common ones.

In the upcoming discussions, we will be discussing about calculating deviations for various types of data. The following diagram represents the methods to calculate the mean deviation from mean for two types of data, i.e. grouped and ungrouped data.

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How to Calculate Mean Absolute Deviation (with Examples)

The mean absolute deviation (mean deviation about the mean) is a measure of the degree of dispersion of data values.

It involves measuring the absolute difference between the data values from the mean and taking the average of these differences. Thus the mean absolute deviation measures the “average” distance of the data values from the mean.

How to Find the Mean Absolute Deviation:

• We first calculate the mean using the formula, X̄ = ∑ x i /n.
• Calculate the differences in the data values from the mean ( x i – X̄ ).
• Take the absolute value of the above differences. This simply means ignoring the sign of the difference if it is negative.
• Take the average of the above absolute values.

Mean Absolute Deviation Formula (Raw and Ungrouped Data):

The mean absolute deviation for raw data can be calculated using the formula,

Mean Absolute Deviation = ∑| x i – X̄ |/n.

If the data is given in ungrouped tabular form with frequencies f i then the mean absolute deviation can be found using the formula,

Mean Absolute Deviation = ∑f i | x i – X̄ |/n.

Consider the data values 2, 3, 7, 8, and 9.

Step 1 : We first obtain the mean as follows, X̄ =(2+3+7+8+9)/5=5.8.

Step 2 : We find the absolute differences in the data values from the mean,

• |2 – 5.8| = |-3.8| = 3.8
• |3 – 5.8| = |-2.8| = 2.8
• |7 – 5.8| = |1.2| = 1.2
• |8 – 5.8| = |2.2| = 2.2
• |9 – 5.8| = |3.2| = 3.2

Step 3 : We now calculate the average of the above absolute values,

Mean Absolute Deviation = (3.8+2.8+1.2+2.2+3.2)/5= 13.2/5 = 2.64.

Mean Absolute Deviation for Grouped Data:

Suppose that the data is given in tabular form with class intervals and corresponding frequencies.

• We first calculate the class mark x i by taking the mid-value for each of the class intervals.
• We then calculate the mean using the formula, X̄ = ∑f i x i /∑f i .
• The mean deviation for grouped data can then be calculated using the formula, Mean Absolute Deviation = ∑f i | x i – X̄|/∑f i .

Consider the following frequency distribution,

Step 1 : We calculate the class mark (mid-value) and the mean of the data.

Mean = ∑f i x i /∑f i = 415/15 = 27.67.

Step 2 : We calculate the absolute differences and take their average.

Mean Absolute Deviation = ∑f i | x i – X̄|/∑f i = 188.01/15 = 12.534.

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Mean Absolute Deviation

By Tina Goosz

Learn about what Mean Absolute Deviation is and how to use it to solve problems. Find the definition, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

Where does this concept fit into the curriculum?

Sample Math Problems

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Practice Math Problems

How can we use the concept: Most often when finding Mean Absolute Deviation (henceforth called MAD), we follow a series of steps and not a formula.

1) Find the mean of the data set.

2) Find the absolute deviation of each data point by finding the absolute value of the difference between each data point and the mean. (Since we are using absolute value, all these values will be positive.)

3) Find the mean of the deviations.

1. Given the data set: 11, 6, 18, 4, 15, 25 find the mean absolute deviation.

Step 1: Find the mean of the data set.

(11 + 6 + 18 + 4 + 15 + 25) 6 = 13.167 (answer rounded to the thousandths place)

Step 2: Find the absolute value of the difference of each data point from the mean.

| 11 - 13.167 | = 2.167

| 6 - 13.167 |= 7.167

| 18 - 13.167 |= 4.833

| 4 - 13.167 | = 9.167

| 15 - 13.167 |= 1.833

| 25 - 13.167 |= 11.833

Step 3: Find the mean of the deviations.

(2.167 + 7.167 + 4.833 + 9.167 + 1.833 + 11.833) 6 = 6.167

2. Given the data set: 30, 49, 17, 17, 46 find the mean absolute deviation.

(30 + 49 + 17 + 17 + 46) 5 = 31.8

| 30 - 31.8 |= 1.8

| 49 - 31.8 |= 17.2

| 17 - 31.8 |= 14.8 We will use this twice in step 3 of solving since this data point occurs twice in the data set.

|46 - 31.8|= 14.2

(1.8 + 17.2 + 14.8 + 14.8 + 14.2) 5 = 12.56

3. Given the data set: 81, 51, 54, 97, 68, 57, 64 find the mean absolute deviation.

(81 + 51 + 54 + 97 + 68 + 64) 6 = 69.167

| 81 - 69.167 | = 11.833

| 51 - 69.167 | = 18.167

| 54 - 69.167 | = 15.167

| 97 - 69.167 |= 27.833

| 68 - 69.167 |= 1.167

| 64 - 69.167 |= 5.167

(11.833 + 18.167 + 15.167 + 27.833 + 1.167 + 5.167) 6 = 13.22 (rounded)

4. Given the data set: 57, 45.3, 58.2, 46.7, 65.5 find the mean absolute deviation.

(57 + 45.3 + 58.2 + 46.7 + 65.5) 5 = 54.54

| 57 - 54.54 |= 2.46

| 45.3 - 54.54 |= 9.24

| 58.2 - 54.54 |= 3.66

| 46.7 - 54.54 |= 7.84

|65.5 - 54.54 |= 10.96

(2.46 + 9.24 + 3.66 + 7.84 + 10.96) 5 = 6.832

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1. Given the data set: 2, 9, 22, 16, 10, 16 find the mean absolute deviation.

2. Given the data set: 12, 38, 37, 25, 12 find the mean absolute deviation.

3. Given the data set: 75, 72, 85, 91, 74, 81 find the mean absolute deviation.

4. Given the data set: 92.4, 90.5, 100.7, 87.7, 93.2 find the mean absolute deviation.

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Mean Absolute Deviation Worksheets

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The concept of mean absolute deviation is an essential topic in the math curriculum and is used extensively in real life as it is precise and accurate. Incorporate these free worksheets on mean absolute deviation and help students understand this concept. It is nothing, but the variability of the given data set and is the average of the absolute deviations from the central point of the data set. Instruct students to find the mean; calculate the difference between each number in the data set and the mean, which is the absolute value; and record the answers in the appropriate columns. Finally find the average of those differences to obtain the mean absolute deviation.

This bundle of printable pdf worksheets is highly recommended for sixth grade and seventh grade students.

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▶ Mean, Median, Mode and Range

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Measures of Variability

A series of free, online video lessons with examples and solutions to help Grade 7 students learn how to informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Related Pages Understanding Variability Sampling Variability Common Core Grade 7 Common Core Mathematics Grade 7 Math

For example , The mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Common Core: 7.SP.3

Suggested Learning Targets {#target)

• I can calculate the mean, range, and the mean absolute deviation (MAD) to compare two data sets (Note: MAD is the average distance between each value and the mean.)
• I can observe the overlap and differences of two data sets with similar variability.
• I can compare two data sets using the range or MAD.

The following diagrams show how to calculate the Mean Absolute Deviation. Scroll down the page for examples and solutions on how to use the Mean Absolute Deviation.

Variability and Deviations from the Mean Summarizing Deviations from the Mean.

• Variability describes how spread out the data is.
• For any given value in a data set, the deviation from the mean is the value minus the mean.
• The greater the variability (spread) of the distribution, the greater the deviations from the mean (ignoring the signs of the deviation).

A consumers' organization is planning a study of the various brands of batteries that are available. As part of its planning, it measures lifetime (how long a battery van be used before it must be replaced) for each of six batteries of Brand A and eight batteries of Brand B. Dot plots showing the battery lives for each brand are shown below. (a) Does one brand of battery tend to last longer or are they roughly the same? Justify your claim. (b) What number could you calculate to compare the typical battery life of the two brands?

Introduction to Mean Absolute Deviation This video explains what Mean Absolute Deviation is as well as how to calculate it.

Example: Find the Mean Absolute Deviation of the data 2, 5, 7, 13, 18

Mean Absolute Deviation (MAD) Review how to find the MAD or mean absolute deviation of a given data set. What is Mean Absolute Deviation (MAD)? Mean absolute deviation is the average distance of all of the elements in a data set from the mean of the same data set.

• The MAD indicates how spread out your data set is.
• A large MAD indicates a data set that is more spread out in relation to the mean.
• A small MAD would indicate data that is less spread out and located closer to the mean.

Example: A student scored the following percentages on 10 quizzes over the course of a semester: 55, 65, 70, 70, 72, 85, 90, 90, 93, 100 Find the MAD of the quiz scores. Step 1: Find the mean of the given data set Step 2: Find the distance that each element in your data set is away from the mean if it were on a number line. (subtract each element with the mean and use the absolute value because the distance is always positive) Step 3: Calculate the mean of all of the values you had when subtracting.

Mean Absolute Deviation This video reviews how to find Mean Absolute Deviation for a set of data. One way to find out how consistent a set of data is to find the Mean Absolute Deviation. The Mean Absolute Deviation describes the average distance from the mean for the numbers in the data set. Step 1: Find the mean of the data. Step 2: Subtract the mean from each data point. (Make all values positive) Step 3: Find the mean of the values you got when you subtracted in step 2.

Example: Find the Mean Absolute Deviation of the following data 87, 94, 72, 65, 97, 77

Measures of Variability Calculating the Range, IQR and MAD or Mean Absolute Deviation for ungrouped data.

Mean Absolute Deviation This video describes how to calculate the mean absolute deviation of a data set with two examples. Steps to find the MAD

• Calculate the mean of the data set.
• Find how far each point is from the mean.
• Take the absolute value of each difference.
• Calculate the mean of the differences.

Example: Find the MAD of each of these data sets. a) 5, 5, 6, 6, 6, 7, 7, 9, 9, 10 b) 1, 1, 2, 2, 3, 4, 7, 10, 12, 28

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Mean Absolute Deviation Calculator

Mean absolute deviation - work with steps.

Mean Absolute Deviation Calculator is an online Probability and Statistics tool for data analysis programmed to calculate the absolute deviation of an element of a data set at a given point. This calculator generate the output values of Mean and Mean Absolute Deviation according to the given input data set

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• Skewness Calculator
• Normalized or Z-Score Calculator
• Standard Error of Mean Calculator
• Margin of Error (ME) Calculator
• Population Confidence Interval Calculator
• Confidence Interval Calculator