Variance and Standard Deviation for Grouped Data

 

 

Variance and standard deviation are statistical measures used to understand the spread or dispersion of data points in a set, whether the data is grouped or ungrouped. However, calculating variance and standard deviation for grouped data involves some modifications to the formulas used for ungrouped data.

Variance for Grouped Data

 When dealing with grouped data, where data points are organized into intervals or classes, we use a slightly different formula to calculate the variance. The formula for variance in grouped data is as follows:

Variance = Σ((f * (x - μ)²)) / N

Where:

• Σ represents the sum of the values.
• f is the frequency (number of observations) in each class.
• x is the midpoint of each class interval.
• μ (mu) is the mean of the data set.
• N is the total number of observations (sum of frequencies).

Standard Deviation for Grouped Data

The formula for standard deviation in grouped data is similar to the formula for variance. We take the square root of the variance to obtain the standard deviation:

Standard Deviation = √(Σ((f * (x - μ)²)) / N)

Let's work through an example to illustrate how to calculate variance and standard deviation for grouped data:

Consider the following grouped data:

Class Intervals

Class Intervals	Frequency (fi)	Xi	fi*xi	Xi- μ	(Xi- μ)2	Fi(Xi- μ)2
10 – 19 -	5	14.5	5*14.5=72.5	14.5-29.5= -15	225	1125
20 – 29	8	24.5	8*24.5=196	24.5-29.5= -5	25	200
30 – 39	12	34.5	12*34.5=414	34.5-29.5= 5	25	300
40 – 49	7	44.5	7*44.5=311.5	44.5-29.5= 15	225	1575
Sum	32		944			3200

 Step 1

 Calculate the midpoint (x) for each class interval. It is the average of the lower and upper limits of the interval.

x1 = (10 + 19) / 2 = 14.5

x2 = (20 + 29) / 2 = 24.5

 x3 = (30 + 39) / 2 = 34.5

x4 = (40 + 49) / 2 = 44.5

Step 2

Calculate the mean (μ) of the data set. It is the weighted average of the midpoints, using the frequencies as weights.

μ = ((x1 * f1) + (x2 * f2) + (x3 * f3) + (x4 * f4)) / N =

((14.5 * 5) + (24.5 * 8) + (34.5 * 12) + (44.5* 7)) / (5 + 8 + 12 + 7)

= 29.5

Step 3

Calculate the squared differences between each midpoint and the mean, weighted by the frequencies. ((f1 * (x1 - μ)²) + (f2 * (x2 - μ)²) + (f3 * (x3 - μ)²) + (f4 * (x4 - μ)²)) / N = ((5 * (14.5 – 29.5)²) + (8 * (24.5 – 29.5)²) + (12 * (34.5 – 29.5)²) + (7 * (44.5 – 29.5)²)) / (5 + 8 + 12 + 7) = 100

Step 4: 

Calculate the variance. Variance= 100

Step 5: 

Calculate the standard deviation. Standard Deviation ≈ √100 ≈ 10

Therefore, for the given grouped data, the variance is approximately 100, and the standard deviation is approximately 10.