Mean deviation for grouped data - examples and formulas

This article will be about The mean deviation for grouped data, with the formula, examples and solved problems.

Calculate the mean deviation for grouped data

The mean deviation is a method that measures the dispersion of the elements of a set respecting to the arithmetic mean. To find this deviation in an ungrouped data is not that complicated, but to calculate the mean absolute deviation in grouped data is a little more complex because we have to do more steps.

Mean deviation for grouped data

To find the mean deviation in grouped data, we also have to know how to find the arithmetic mean for grouped data, because the formula to find the mean deviation also uses the arithmetic mean, although there are some cases where the arithmetic mean is already given in the problem.

The mean deviation can be found by introducing directly the data in the formula, but this would not be that ordered, this is why in the examples of this article we are going to solve every part of the formula in tables and then we are going to take each part of the formula and solve the problem.

Formula mean deviation for grouped data

The mean absolute deviation formula is only one, but down below we are presenting two, the first is for the arithmetic mean for grouped data and the second is for the mean deviation, because, as we said, we have to find the arithmetic mean first, to just then find the mean deviation.

  • Arithmetic mean formula
  • x =
    Σ x * f/n
  • Mean deviation formula
  • dm =
    Σ |x - x| * f/n

Examples of mean deviation for grouped data

Example 1: The ages of a group of people was taken and the compiled data is in the next table below.

This problem will be solved step by step so you can understand how to find every variable, first we are going to find a column and then we are going to explain what is the column for.

f: the letter f represents the frequency that every interval has, or in this case, how many people is in the interval of each group of age.

ages f
]20-30] 8
]30-40] 10
]40-50] 5

x: is the middle value of each interval, it is calculated by adding the low limit plus the high limit and then we divide the result by 2.

  • Example
  • x de ]20-30] =
    20 + 30/2
  • x de ]20-30] =
    50/2
  • x de ]20-30] = 25
ages f x
]20-30] 8 25
]30-40] 10 35
]40-50] 5 45
Total (Σ) 23

f * x: is the multiplication of the frequency of each interval by the middle number of each interval, this column is use to find the mean.

ages f x f * x
]20-30] 8 25 200
]30-40] 10 35 350
]40-50] 5 45 225
Total (Σ) 23 775
  • With these data we can find the mean
  • x =
    Σ x * f/n
  • x =
    775/23
  • x = 33.69

|x - x|: is the absolute value of the subtraction of x minus the mean.

ages f x f * x |x - x|
]20-30] 8 25 200 8.7
]30-40] 10 35 350 1.3
]40-50] 5 45 225 11.3
Total (Σ) 23 775

|x - x| * f: And last we calculate the last column by multiplying the previous column by the frequency of every interval.

ages f x f * x |x - x| |x - x| * f
]20-30] 8 25 200 8.7 69.6
]30-40] 10 35 350 1.3 13
]40-50] 5 45 225 11.3 856.5
Total (Σ) 23 775 1391.1

And finally we calculate the mean deviation.

  • dm =
    Σ |x - x| * f/n
  • dm =
    1391.1/23
  • dm = 6.05

Example 2: A group of people was asked about the average meal time (in minutes), calculate the mean deviation of the average time of the meal of the people.

minutes f
]5-15] 5
]15-25] 30
]25-35] 22
]35-45] 10
Total (Σ) 67

With these results, first we are going to find the columns to calculate the mean of the data, that are “x” and “x*f” (If your forgot how to find them go check the previous example)

minutes f x x*f
]5-15] 5 10 50
]15-25] 30 20 600
]25-35] 22 30 660
]35-45] 10 40 400
Total (Σ) 67 1710
  • With this data we can find the mean.
  • x =
    Σ x * f/n
  • x =
    1 710/67
  • x = 25.52

And now we find the |x - x| column.

minutes f x x*f |x-x| |x-x| * f
]5-15] 5 10 50 15.53 77.6
]15-25] 30 20 600 5.52 165.6
]25-35] 22 30 660 4.48 98.56
]35-45] 10 40 400 14.48 144.8
Total (Σ) 67 1710 486.56

And now we can calculate the mean deviation.

  • dm =
    Σ |x - x| * f/n
  • dm =
    486.56/67
  • dm = 7.26

Related articles

Mean deviation Mean median and mode Mean for grouped Median for grouped Mode for grouped Quartiles Standard deviation Frequency table