The variance and the standard deviation both are two useful and crucial statistical terms that are interlinked with one another. These terms are important when measuring the statistical values dispersion. A distribution’s dispersion is the amount by which its values vary from the distribution’s average. The amount of variance can be measured using a variety of metrics. The degree of dispersion is determined by measuring the variation of data points.
The terms variance and standard deviation are related because computing the square root of the variance provides the square root of the standard deviation for the specified data values. Meaningful measures of the variability of the data values are the variance, mean, and standard deviation. Measures of variance and standard deviation indicate how much and how far apart the data points are from the mean, respectively.
Two metrics are used to assess an investment’s risk: variance and standard deviation. The risk associated with investments rises as variance or standard deviation increases. Because the return on investment is determined using the mean, it is also significant. In this article, we will elaborate on the terms of the standard deviation and the variance in detail.
What is Variance?
The squared difference between the mean and each observation is the variance of these given data values. In 1918, R.A. Fisher delivered the idea of variance. Because of its significance, variance is widely used for measuring dispersion. Mathematically:
- Sk 2 = ? (x – x?)2 / n where Sk 2 is variance and ? is the summation sign.
Or
- Sk 2 = (? x2 / n) – (?x / n)2
A set of data’s variance describes the degree of dispersion within it. The above-mentioned relationships indicate that as the values given approach one another and become equivalent, the variance decreases to zero. Any values that are not zero have positive variances.
When the data points are far separated from the mean and each other, it indicates a significant variation; when the data points are close to the mean and each other, it indicates minimal variance.
Defining the Standard Deviation (SD):
The positive square root of the variance signifies the important term, standard deviation. To compute the value of the standard deviation, three variables are required. In a data collection, the value of each point represents the initial variable, and a sum number represents each subsequent variable (x, x1, x2, x3, etc.)
The mean is then applied to the values of the variables x and n, as well as the data values given to them. Symbolically
- Sk = ? [? (x – x?)2 / n] where Sk is the standard deviation.
Or
- Sk = ? [(? x2 / n) – (?x / n)2]
The units used to represent the standard deviation match or correspond to the units used to represent the observations. One metric that shows how different something is from the mean is the standard deviation, often known as dispersion or spread. The standard deviation helps visualize a typical variation from the mean.
It is a preferred measure of variability since it goes back to the original units of data values of measurement. Just like with variance, there is a big variation if the data points are widely scattered from the mean and a small variation if the data points are near the mean.
The variance that the numbers depart from the average is determined by the standard deviation. Using the standard deviation, which is based on all data, is the simple method of evaluating dispersion. Consequently, a slight variation in one data point has an impact on the standard deviation.
How to calculate standard deviation and variance?
Using online tools is an easy way to calculate standard deviation and variance. Below are a few solved examples for calculating standard deviation and variance manually.
Example 1:
Calculate what will be the variance and the standard deviation for the following given scores of the students in the table.
| Student | Anas | Maha | Moiz | Ali | Saim | Syam | Fiaz | Sami | Umer |
| Score (xi) | 78 | 61 | 82 | 59 | 32 | 93 | 44 | 26 | 23 |
Solution:
Step 1: Now we will perform the following necessary computations as given in the table:
| xi | 78 | 61 | 82 | 59 | 32 | 93 | 44 | 26 | 23 | ?X = 498 |
| xi 2 | 6084 | 3721 | 6724 | 3481 | 1024 | 8649 | 1936 | 676 | 529 | ?X2 = 32824 |
Step 2: We will apply the relevant formula according to the computations that we perform in the above table.
The formula for variance:
Sk 2 = (? x2 / n) – (?x / n)2
Putting the relevant values:
Sk 2 = (32824 / 9) – (498 / 9) ^ 2
S2 = 3647.11 – (248004 / 81)
Sk 2 = 3647.11 – 3061.78
Sk 2 = 585.33 scores ^ 2 Ans.
For standard deviation:
Sk = ? [(? x2 / n) – (?x / n) ^ 2]
Sk = ? (585.33)
Sk = 24.19 scores Ans.
Example 2:
Compute what will be the variance and the standard deviation for the values given in the following table.
| 3 | 6 | 7 | 8 | 10 | 12 | 12 | 14 |
Solution:
Step 1: First of all, we are to compute the average of the given data values in the above table.
x? = (3 + 6 + 7 + 8 + 10 + 12 + 12 + 14) / 8
x? = (72) /8
x? = 9
Step 2: Now we are to compute the following necessary computations in the table to proceed to the next for determining the variance and the standard deviation.
| x | (xk – x?) | (xk – x?)2 |
| 3 | -6 | 36 |
| 6 | -3 | 9 |
| 7 | -2 | 4 |
| 8 | -1 | 1 |
| 10 | 1 | 1 |
| 12 | 3 | 9 |
| 12 | 3 | 9 |
| 14 | 5 | 25 |
| Total | ? (x – ?)2 = 94 |
Step 3: We will apply the relevant formula according to the computations that we perform in the above table.
The formula for variance:
Sk 2 = ? (xk – x?) 2 / n
Putting the relevant values:
Sk 2 = (94) / 8
Sk 2 = 11.75
For standard deviation:
Sk = ? [? (x – x?)2 / n]
Sk = ? (11.75)
Sk = 3.4278 Ans.
Wrap Up:
In this article, we have covered the key ideas of variance and standard deviation in a lot of detail. We have discussed the meanings of these terms as well as the numerous mathematical relations that enable us to compute these crucial terms for observation and analysis of the given data.