Like standard deviation, we can also find variance for two different types of datasets based on the sample size. They are called population variance and sample variance.
We use population variance when we take all of the data in the dataset under consideration, whereas we use sample variance when we consider only a subset of the total population.
Let us now discuss them in detail.
Population Variance
Population variance measures the dispersion of data points across an entire population. It is represented by the Greek letter sigma squared (σ²)
In statistics, data can be ungrouped (raw) or grouped data (well-organized). We can calculate the variance for each type of data.
For Ungrouped Data
Mathematically, the formula to find the population variance is:
Thus, the population variance of the grouped data is 132.24
Sample Variance
Sample variance measures variability when the data represents a subset (sample) of the total population. To avoid any biases, we use the correction factor, n – 1, known as Bessel’s correction.
The sample variance is represented by the letter s²
Note: This adjustment improves the accuracy by correcting the variability in small samples and making the sample variance a better approximation of the population variance.
For Ungrouped Data
Mathematically, sample variance can be obtained using the formula:
Thus, the sample variance of the grouped data is 136.42
Note: We observe that the sample variance is greater than the population variance for each dataset for both grouped and ungrouped data.
Properties
Non-Negativity
Variance is always non-negative because it is the average of squared deviations. Mathematically:
σ2 ≥ 0
s2 ≥ 0
Zero Variance
If all sample data points in a population or a sample are identical, the variance equals 0. It means
σ2 = 0 when xi = μ, ∀ i
s2 = 0 when xi = ${\overline{x}}$, ∀ i
Units
Variance is measured in squared units of the data. For example, if data is measured in meters, the population and sample variances are in square meters.
Adding a Constant
If a constant c is added to all data points, the population and sample variance remain unchanged. It means
Var(xi + c) = Var(xi)
Multiplying by a constant
If all data points are multiplied by a constant c, the population and sample variances are scaled by c2:
Var(c ⋅ xi) = c2 ⋅ Var(xi)
Additivity
For independent random variables X and Y, the variance of their sum is:
Var(X + Y) = Var(X) + Var(Y)
Solved Examples
A company measures the heights (in cm) of 5 employees in a department. The data is as follows: 150, 160, 170, 180, 190. Find the population variance.
Solution:
Here, Mean = μ = ${\dfrac{150+160+170+180+190}{5}}$ = 170 The square differences = (xi – μ)2 (150 – 170)2 = (-20)2 = 400 (160 – 170)2 = (-10)2 = 100 (170 – 170)2 = (0)2 = 0 (180 – 170)2 = (10)2 = 100 (190 – 170)2 = (20)2 = 400 The sum of the square differences = ${\sum \left( x_{i}-\mu \right) ^{2}}$ = 400 + 100 + 0 + 100 + 400 = 1000 As we know, the population variance is σ² = ${\sigma ^{2}=\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}$ Now, by using the formula, we get = ${\dfrac{1000}{5}}$ = 200 Thus, the population variance is 200 cm².
A researcher randomly selects 4 students’ scores from a class: 12, 14, 16, 18. Find the sample variance.
Solution:
Here, Mean = ${\overline{x}}$ = ${\dfrac{12+14+16+18}{4}}$ = 15 The square differences = ${\left( x_{i}-\overline{x}\right) ^{2}}$ (12 – 15)2 = (-3)2 = 9 (14 – 15)2 = (-1)2 = 1 (16 – 15)2 = (1)2 = 1 (18 – 15)2 = (3)2 = 9 The sum of the square differences = ${\sum \left( x_{i}-\overline{x}\right) ^{2}}$ = 9 + 1 + 1 + 9 = 20 As we know, the sample variance is ${s^{2}=\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}$ Now, by using the formula, we get = ${\dfrac{20}{4-1}}$ ≈ 6.67 Thus, the sample variance is 6.67.
