Statistics refers to the collection, analysis, interpretation and presentation of masses of numerical data. Variance and the standard deviation are important topics in statistics. Variance is a measure of central dispersion. It is the average of the squared difference from the mean. Standard deviation calculates the dispersion of a dataset relative to its mean. A standard deviation is a useful tool in investing and trading strategies. If the data points are away from the mean, there is a higher deviation within the data set. Hence, the more spread out the data, the greater the standard deviation. The mean is the average of a given set of observations. In this article, we will discuss how to find the variance from the standard deviation.
What is Variance and Standard Deviation?
The average of the squared differences from the mean is known as the variance.
Variance is denoted by σ2. The square root of the σ2 gives the standard deviation. Standard deviation calculates the dispersion of a dataset relative to its mean.
How to Calculate Variance from Standard Deviation?
You can calculate the variance from standard deviation in a single step.
If standard deviation(σ) is given, then find the square of σ.
σ2 gives the variance.
Similarly, we can find the standard deviation from the variance.
Formulas of Variance and Standard Deviation
| Variance | Standard deviation | |
| Sample | \(\begin{array}{l}s^2=\frac{1}{n-1}\sum _{i=1}^{n}(x_i-\bar{x})^2\end{array} \) |
\(\begin{array}{l}s=\sqrt{\frac{1}{n-1}\sum _{i=1}^{n}(x_i-\bar{x})^2}\end{array} \) |
| Population | \(\begin{array}{l}\sigma^2 =\frac{1}{N}\sum _{i=1}^{n}(x_i-\mu)^2\end{array} \) |
\(\begin{array}{l}\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^{n}(x_i-\mu)^2}\end{array} \) |
Solved Examples
Example 1: The standard deviation of a set of numbers is 6.2. The variance is given by
(1) 38.04
(2) 2.48
(3) 38. 44
(4) 2.08
Solution:
Given standard deviation, σ = 6.2
So variance = σ2 = 6.22 = 38.44
Hence option (3) is the answer.
Example 2: Calculate the variance from the standard deviation of the data set: 18, 22, 19, 25, 12
Solution:
Let xi: = 18, 22, 19, 25, 12
Here, n = 5
Mean (x̄) = (18 + 22 + 19 + 25 + 12)/5
= 96/5
= 19.2
Let us calculate the value of deviations, i.e., (xi – x̄).
(xi – x̄) = (18 – 19.2), (22 – 19.2), (19 – 19.2), (25 – 19.2), (12 – 19.2)
= -1.2, 2.8, -0.2, 5.8, -7.2
Now, square the deviations.
(xi – x̄)2 = (-1.2)2, (2.8)2, (-0.2)2, (5.8)2, (-7.2)2
= 1.44, 7.84, 0.04, 33.64, 51.84
∑(xi – x̄)2 = 1.44 + 7.84 + 0.04 + 33.64 + 51.84
= 94.80
(1/n-1)∑(xi – x̄)2 = 94.80/4 = 23.7
Standard deviation
= √23.7
= 4.9 (approx)
Variance
= 23.7
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Frequently Asked Questions
What do you mean by variance in statistics?
The average of the squared differences from the mean is called variance. Variance is denoted by σ2.
What do you mean by standard deviation?
A standard deviation is a measure of how dispersed the data is in relation to the mean. It is denoted by σ.
Give the formula for finding the variance.
Variance is given by σ2 = ∑(xi-x̄)2/N
x1, x2, …., xN are the N observations.
How do you find the variance if the standard deviation is given?
The square of the standard deviation gives the variance.
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