Average Absolute Deviation Formula

Average absolute deviation is defined as the mean of absolute values of the deviations from the data set values. It is also known as the mean absolute deviation. It is one of the measures of central tendency among mean. median, mode and standard deviations. It provides more accuracy and is easy to use in real-life situations. Its formula equals the sum of differences between the mean and data values divided by the number of values in data set. Its value is always greater than zero as it takes the modulus of differences. 

Formula

M = Σ (x_i – x̄)/n

where,

M is the average absolute deviation,

x̄ is the mean of data set,

Σ (x_i – x̄) is the summation of deviations from mean,

n is the number of values in data set.

Sample Problems

Problem 1. Calculate the average absolute deviation of the data set, 2, 6, 7, 4, 1.

Solution:

The data set is 2, 6, 7, 4, 1.

Here, n = 5.

Mean of the data, x̄ = (2 + 6 + 7 + 4 + 1)/5

= 20/5

= 4

Using the formula we get,

M = Σ (x_i – x̄)/n

= [|4 – 2| + |4 – 6| + |4 – 7| + |4 – 4| + |4 – 1|]/5

= (2 + 2 + 3 + 0 + 3)/5

= 10/5

= 2

Problem 2. Calculate the average absolute deviation of the data set, 5, 8, 3, 0, 1, 7.

Solution:

The data set is 5, 8, 3, 0, 1, 7.

Here, n = 6.

Mean of the data, x̄ = (5 + 8 + 3 + 0 + 1 + 7)/6

= 24/6

= 4

Using the formula we get,

M = Σ (x_i – x̄)/n

= [|4 – 5| + |4 – 8| + |4 – 3| + |4 – 0| + |4 – 1| + |4 – 7|]/6

= (1 + 4 + 1 + 4 + 3 + 3)/6

= 16/6

= 2.67

Problem 3. Calculate the average absolute deviation of the data set, 6, 9, 4, 2, 1, 8.

Solution:

The data set is 6, 9, 4, 2, 1, 8.

Here, n = 6.

Mean of the data, x̄ = (6 + 9 + 4 + 2 + 1 + 8)/6

= 30/6

= 5

Using the formula we get,

M = Σ (x_i – x̄)/n

= [|5 – 6| + |5 – 9| + |5 – 4| + |5 – 2| + |5 – 1| + |5 – 8|]/6

= (1 + 4 + 1 + 3 + 4 + 3)/6

= 16/6

= 2.67

Problem 4. Calculate the sum of deviations from the mean if the average absolute deviation is 15 and the number of observations is 6.

Solution:

We have,

M = 15

n = 6

Using the formula we get,

M = S/n

=> S = Mn

= 15 (6)

= 90

Problem 5. Calculate the sum of deviations from the mean if the average absolute deviation is 12 and the number of observations is 9.

Solution:

We have,

M = 12

n = 9

Using the formula we get,

M = S/n

=> S = Mn

= 12 (9)

= 108

Problem 6. Calculate the number of observations if the sum of deviations from the mean is 336 and the average absolute deviation is 12.

Solution:

We have,

S = 336

M = 12

Using the formula we get,

M = S/n

=> n = S/M

= 336/12

= 28

Problem 7. Calculate the number of observations if the sum of deviations from the mean is 518 and the average absolute deviation is 14.

Solution:

We have,

S = 518

M = 14

Using the formula we get,

M = S/n

=> n = S/M

= 518/14

= 37

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