Mean Absolute Deviation

The measure of spread represents the amount of dispersion in a data-set. i.e how spread-out are the values of data-set around the central value(example- mean/mode/median).It tells how far away the data points tend to fall from the central value.

• The lower value of the measure of spread reflects that the data points are close to the central value. In this case, the values in a data-set are more consistent.
• Further, the distance of the data points from the central-value, the greater is the spread. whereas here, the values are not much consistent.

Distribution of Data

Using the above diagram, we can infer that the narrow distribution represents a lower spread, and the broad distribution represents a higher spread.

Range

The range is the simplest measure of variation. It is defined and calculated as the difference between the largest and smallest values of the data-set.

Range = largest value – smallest value

• A small value of range means the data is quite consistent and most of the data-points lie near to the mean.
• Whereas a higher range means the data is quite inconsistent and the data has extreme values, and the data-points don’t lie near to the mean.
• The range doesn’t consider each value of the dataset. Thus, it gives a rough idea of the data-set and its variability.

Examples

Example 1: The data given are: 8, 10, 4, 1, 15. Calculate the range of the given data?

Solution: 

The data in ascending order is = 1  4  8  10  15.

Range = largest – smallest

           = 15 – 1

Range = 14

Example 2: What is the range of these integers?

14, -18, 7, 0, -5, -8, 15, -10, 20

Solution: 

The data in ascending order is =  -18, -10, -8, -5, 0, 7, 14, 15, 20

Range = largest – smallest

           = 20 – (-18)

Range = 38

Example 3: Calculate the range of the given data:

8, 10, 5 , 14 , 42, 3566

Solution: 

The data in ascending order is = 5, 8, 10, 14, 42, 3566

Range = largest – smallest

           = 3566 – 5

Range = 3561

Mid-Range

The mid-range is the value midway between the largest and smallest value of a data-set. It is calculated as the mean of the largest value and smallest value of the data-set.

Mid-Range = (largest value + smallest value)/2

Examples

Example 1: The data given is 8, 10, 5, 9, 11. Calculate the mid-range of the given data?

Solution: 

The data in ascending order is = 5 8 9 10 11

Mid-Range = (largest value + smallest value)/2

                   = (5 + 11)/2

                   = 16/2

mid-range = 8

Example 2: You take 7 statistics tests over the course of a semester. You score 94, 88, 74, 84, 91, 87 and 79. What is the mid-range of your scores?

Solution:

The scores in ascending order is = 74 79 84 87 88 91 94

Mid-Range = (largest value + smallest value)/2

                   = (94 + 74)/2

                   = 168/2

mid-range = 84

Example 3: The height of 8 students in centimeters is given as  120, 132, 117, 126, 110, 135, 150, and 143. Calculate the mid-range of the given data?

Solution:

The scores in ascending order is = 110 117 120 126 132 135 143 150 

Mid-Range = (largest value + smallest value)/2

                   = (150 + 110)/2

                   = 260/2

mid-range = 130

Mean Absolute Deviation (MAD)

The mean absolute deviation (MAD) of a data-set is the average distance between each data point of the data-set and the mean of data. i.e it represents the amount of variation that occurs around the mean value in the data-set. It is also a measure of variation. It is calculated as the average of the sum of the absolute difference between each value of the data-set and the mean. 

MAD = (∑ |xi – mean| ) ÷ n

where 1 < i < n and n is the number of data-points in the data-set.

Examples

Example 1: The data-set is 11 , 15 , 18 , 17 , 12 , 17. Calculate the mean absolute deviation of the given data-set?

Solution:

Step 1: Calculating the mean 

x̅ =  (x₁ + x₂ + x₃ + …… + x_n) / n

x̅ = (11 + 15 + 18 + 17 + 12 + 17 ) / 6 

x̅ = 15 

The mean of the given data = 15 

Step 2: Calculating the absolute difference between each data-point and mean.

Data-Point	Absolute Difference from mean
11	|11 – 15| = 4
12	|12 – 15| = 3
15	|15 – 15| = 0
17	|17 – 15| = 2
17	|17 – 15| = 2
18	|18 – 15| = 3

Step 3: Adding the Absolute Difference together

(∑ |x_i – mean| ) = 4 + 3 + 0 + 2 + 2 + 3 

(∑ |x_i – mean| ) = 14 

Step 4: Dividing the sum of absolute difference and the number of data-points.

MAD =  (∑ |x_i – mean|) ÷ n 

MAD = 14/6 

MAD = 2.33

Hence, we can conclude that, on average, each data-point is 2 distance away from the mean. 

Example 2: The following table shows the number of oranges that grew on Nancy’s orange tree each season

Season	Number of Oranges
Winter	5
Summer	17
Spring	24
Fall	10

Find the mean absolute deviation (MAD) of the data set?

Solution:

Step 1: Calculating the mean

x̅ =  (x1 + x2 + x3 + …… + xn) / n

x̅ = (5 + 17 + 24 + 10) / 4

x̅ = 56/4

The mean of the given data = 14

Step 2: Calculating the absolute difference between each data-point and mean

Data-Point	Absolute Difference from mean
5	|5 – 14| = 9
17	|17 – 14| = 3
24	|24 – 14| = 10
10	|10 – 14| = 4

Step 3:Adding the Absolute Difference together

(∑ |xi – mean| ) = 9 + 3 + 10 + 4 

(∑ |xi – mean| ) = 26

Step 4: Dividing the sum of absolute difference and the number of data-points

MAD =  (∑ |x_i – mean| ) ÷ n

MAD = 26 / 4

MAD = 6.5

Example 3: Consider the following data-set

Name of the student	Marks in Maths
Chetan	90
Shubham	74
Riya	80
Manu	92

Calculate the mean absolute deviation of the given data?

Solution:

Step 1: Calculating the mean

x̅ =  (x1 + x2 + x3 + …… + xn) / n

x̅ = (90 + 74 + 80 + 92) / 4

x̅ = 336/4

The mean of the given data = 84

Step 2: Calculating the absolute difference between each data-point and mean

Data-Point	Absolute Difference from mean
90	|90 – 84| = 6
74	|74 – 84| = 10
80	|80 – 84| = 4
92	|92 – 84| = 8

Step 3: Adding the Absolute Difference together

(∑ |xi – mean| ) = 6 + 10 + 4 + 8

(∑ |xi – mean| ) = 28

Step 4: Dividing the sum of absolute difference and the number of data-points

MAD =  (∑ |xi – mean|) ÷ n

MAD = 28 / 4

MAD = 7