Mean, Variance and Standard Deviation

Mean, Variance, and Standard Deviation are vital statistical measures. Variance quantifies data point deviation from the mean, while Standard Deviation gauges data distribution. The key distinction lies in Standard Deviation being in the same units as the mean, whereas Variance is in squared units. Dive deeper into these concepts with definitions, formulas, and an illustrative example.

Mean

Mean is average of a given set of data. Let us consider below example  

These eight data points have the mean (average) of 5: 





Where ? is mean and x₁, x₂, x₃…., x_i are elements.Also note that mean is sometimes denoted by 

Variance

Variance is the sum of squares of differences between all numbers and means. 
Deviation for above example. First, calculate the deviations of each data point from the mean, and square the result of each: 



variance = = 4. 



Where ? is Mean, N is the total number of elements or frequency of distribution.

Standard Deviation

Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean. 

Standard Deviation (for above data) = = 2 

Why did mathematicians choose a square and then square root to find deviation, why not simply take the difference of values? 
One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems. [Source: MIT Video Lecture at 1:19] 

1. Value of standard deviation is 0 if all entries in input are same.
2. If we add (or subtract) a number say 7 to all values in the input set, the mean is increased (or decreased) by 7, but the standard deviation doesn’t change.
3. If we multiply all values in the input set by a number 7, both mean and the standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, the mean is multiplied by -7, but the standard deviation is multiplied by 7.
4. Standard deviation and variance is a measure that tells how spread out the numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
5. Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).

Check:

• Variance and Standard Deviation
• Real Life Applications of Standard Deviation
• Difference Between Variance and Standard Deviation

Below questions have been asked in previous year GATE exams 
https://www.geeksforgeeks.org/gate-gate-cs-2012-question-64/amp/

References:
https://en.wikipedia.org/wiki/Standard_deviation
http://staff.argyll.epsb.ca/jreed/math30p/statistics/standardDeviation.htm

Mean, Variance and Standard Deviation – FAQs

What is the Difference Between Standard Deviation and Variance?

Standard deviation and variance both measure the spread of data points in a dataset relative to the mean. The key difference is that variance measures the average of the squared deviations from the mean, while standard deviation is the square root of the variance, providing a measure of spread in the same units as the data.

How Do You Calculate Mean, Variance, and Standard Deviation?

• Mean: Add all the numbers together and divide by the count of numbers.
• Variance: Calculate the mean, subtract the mean from each number, square the result, sum these squared results, and divide by the count of numbers minus one.
• Standard Deviation: Take the square root of the variance.

Why Are Mean, Variance, and Standard Deviation Important?

These statistical measures are crucial for understanding the distribution of data. The mean provides a central value, while variance and standard deviation give insights into the data’s variability or spread, indicating the consistency or volatility of the dataset.

Can Variance and Standard Deviation Be Negative?

No, variance and standard deviation cannot be negative. Variance is calculated as the average of the squared differences from the Mean, resulting in a non-negative value. Since standard deviation is the square root of variance, it also cannot be negative.

How Does Outliers Affect Mean, Variance, and Standard Deviation?

Outliers can significantly affect the mean by pulling it towards the outlier value, thus not accurately reflecting the dataset’s central tendency. Variance and standard deviation are also affected as they will increase, indicating a higher spread of data due to the outlier(s).