Mean And Variance

Mean and variance are important measures in statistics. The mean gives the average of a set of numbers. Variance shows how far the numbers spread out from the mean. Let us understand them in detail.

What is Mean?

Mean measures the average of a group of values. It is computed by dividing the sum of observations by the total number of observations.

The four types of means are:

Arithmetic
Geometric
Harmonic
Weighted Arithmetic

A disadvantage of mean as a central tendency metric is that it is very open to outliers. The data set may have values that are significantly greater or less than the rest of the data.

For an ungrouped dataset

formula

For grouped datasets or discrete frequency data

The formula is

formula

Where

x̄ = mean

Σfx = summation of the product of frequency and class

Σf = summation of the frequencies

Example: Find the mean weight from the following data.

Age	Frequency
10	9
11	8
12	10
13	7
14	4

Solution:

formula

The formula for the mean of grouped data is given as

formula

Properties of Mean

The mean has the following properties:

When each value is increased by k, the mean also increases by k.
When each value is decreased by k, the mean also decreases by k.
If every value is multiplied by k, the mean is also multiplied by k.
If every value is divided by k, the mean is also divided by k.

What is Variance?

Variance is a measure of dispersion. It helps assess how data deviates from its average value. In other words, it is the average of the squared differences between each value and the mean.

Data can be either grouped or ungrouped. Grouped data is organised into class intervals, while ungrouped data comprises individual data points.

Specific Symbols for Variance

σ²: (sigma squared): Denotes the variance of an entire population.
s²: Denotes the variance of a sample taken from a population.
Var(X): A general notation for the variance of a random variable X.

Statistical variance is put in two categories depending on your dataset:

Population variance
Sample variance

For an ungrouped dataset

The formulas for population variance and sample variance are as follows:

Population Variance:

formula

Here,

σ² represents the population variance

xᵢ represents each observation in the data

μ is the population mean

N is the total number of observations or the population size.

Sample Variance:

formula

Here,

s² is the sample variance

xᵢ represents individual data points

x̄ is the sample mean

n is the sample size or the number of data values in the sample

Example: Calculate the Variance of Test Scores

Suppose four students scored: 70, 75, 80, and 85. To calculate the variance:

Step 1: Calculate the mean (average) of the scores:

formula

Step 2: Calculate the squared differences from the mean:

(70 − 77.5)² = 56.25

(75 − 77.5)² = 6.25

(80 − 77.5)² = 6.25

(85 − 77.5)² = 56.25

Step 3: Take the average of the squared differences:

formula

For a grouped dataset

The variance formulas are:

Population Variance for Grouped Data:

formula

Here,

f represents the frequency of each class

x is the midpoint of the class

Sample Variance for Grouped Data:

formula

Example: The marks obtained by 20 students in a test are grouped as follows:

Marks (Class Interval)	Frequency (f)
0–10	2
10–20	3
20–30	5
30–40	6
40–50	4

Step 1: Find the midpoints (x) of each class.

Class Interval	Frequency (f)	Midpoint (x)
0–10	2	5
10–20	3	15
20–30	5	25
30–40	6	35
40–50	4	45

Step 2: Compute f×x

formula

Step 3: Find squared deviations (x − x̄)² and f(x − x̄)²

x	f	x − 30.5	(x − 30.5)²	f(x − 30.5)²
5	2	-25.5	650.25	1300.5
15	3	-15.5	240.25	720.75
25	5	-5.5	30.25	151.25
35	6	4.5	20.25	121.5
45	4	14.5	210.25	841.0

formula

Properties of Variance

If the variance is zero, it means all values in the dataset are the same as the mean.
A small variance shows that the data points are close to the mean, while a large variance shows that the data is widely spread out.
If we add or subtract a constant (a) to every observation, the variance does not change.
If we multiply every observation by a constant (a), the variance gets multiplied by a².

Summary

Mean and variance in statistical calculations are important as they describe central dispersion in a dataset. Mean represents the average (central tendency). On the other hand, the variance quantifies the spread or uncertainty around that average.