Mathematics

# How to Calculate Variance of Frequency Data

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Variance is a term which indicates the variation of a given set of statistical data from the measured value. Mathematically, the square root of standard deviation is called variance, denoted by σ2.

In statistics, there are two type of data distribution. 1.Raw data (Ungrouped) 2. Grouped data.

A: Method to calculate variance of ungrouped data (raw data)

Step 1. First of all, find the mean mean(average) of the raw data.

Step 2. Now find the sum of square of all the numbers

Step 3. Now divide the result from step 2 (sum of square of all the numbers) by the total number of variables (frequency)

Step 4. Now apply :  Result of step 3 - square of average value obtained in step 1

Step 4. Now Apply the square root of the result obtained in step 4

Example:

Let us consider data : 3,5,10,14,18,22

Step 1.  Its average =(3+5+10+14+18+22)/6=12

Step 2. Sum of squares: 32+52+102+142+182+222 =1138

Step 3. Divide sum of squares by frequency: 1138/6 = 189.67

Step 4. Apply  result of step 3-square of average: 189.67-122 = 189.67-144 = 45.67

Step 5. Now apply square root of result of step 4: = 6.757 = 6.78

Hence variance = 6.78

B: Method to calculate variance of grouped data (frequency distribution): in case of frequency distribution there are two types .1. Tabulated frequency distribution 2. Untabulated frequency distribution

In tabulated frequency distribution, class intervals and their corresponding frequency are given, but in the case of untabulated frequency distribution, variables with their corresponding frequencies are given. In the case of untabulated frequency distribution, variable is denoted by x, whereas in case of tabulated frequency distribution, class-mark is taken as x.

Step1. Find Σf = n (sum of frequency)

Step 2. Now multiply frequency of each class (or variable) with the corresponding value of x , say it Σfx.

Step3. Find mean value as Σfx /Σf =m(say)

Step 4. Now subtract mean value from each of the class values x as (x-m)

Step 5. Now square , each value of (x-m) of each class as (x-m)2

Step 6. Now multiply each values of (x-m)2 with the corresponding frequency f as f.(x-m)2

Step 7. Now sum each values of f.(x-m)2 as Σf.(x-m)2

Step 8. Now Apply variance = Σf.(x-m)2/n and get the result.

Example: Find variance of the following frequency distribution:

x: 4   6 9 11 15 18 20 23 25

f:  2   4 6 5 8 7 4 9 5

Ans:

x          :4          6 9 11 15 18 20 23 25        Total

f           :2        4   6 5 8 7 4 9 5         50

fx         :8     24 54 55 120 126 80 207 125      799

(x-m)    : -11.98    -9.98 -6.98 -4.98 -0.98 2.02 4.02 7.02 9.02

(x-m)2  :143.52     99.60 48.72 24.80 0.96 4.08 16.16 49.28 81.36

f (x-m)2:287.04      398.40   292.32 124.00 7.68 28.56 64.64 443.52      406.80 2052.96

Here Σf = 50 and Σfx = 799

Now Mean m= Σfx /Σf = 799/50 = 15.98

Here we have got Σf (x-m)2 = 2052.96

Hence, Variance of the frequency distribution is σ2 = Σf (x-m)2/Σf = 2052.96/50 = 41.0592 = 41.01

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