Variance and Standard deviation: Overview, Questions, Preparation

Standard deviation and variance are concepts of statistics that are calculated with the help of the mean or average. These concepts are used in various fields, such as accounting, economics, and even in the financial sector. 

Variance

The variance is the average of the squared differences from the mean. 
Formula-

xi = the ‘i’ th data point
x=the mean of all data points
n=the number of data points

Standard deviation is calculated from the derived variance as mentioned above. The square root of the variance generates the standard deviation. 
Formula-

The topic is introduced for the first time to the students in Grade 11 as a part of the chapter Statistics. Students can expect questions based on this concept in various competitive examinations. 

1. Find the mean and standard deviation using the short-cut method.

Solution.

Mean (x)=a + ∑ fidi/ ∑fi = 65 +(-100)/100 = 65-1 = 64

Variance (σ²) = 1/N∑ fi(xi-x)² = 1/100 ×286 = 2.86

Standard deviation = √ Variance = √ 2.86 = 1.69

2. Find the mean and variance for the first ‘n’ natural numbers.

Solution.

Mean (x) = Sum of all observations/ Number of observations
Thus,  (x)= [n(n+1)/2] / n = n+1/2
Variance (σ²) = 1/N∑ fi(xi-x)²
= 1/n∑ [xi(n+1/2)]² 
=1/n[n(n+1)(2n+1)/6 - (n+1/n)[n(n+1)/2] +(n+1)²/4n×n
=[(n+1)(n-1)]/12
=[n² - 1]/12

3. Find the mean, variance, and standard deviation using the short-cut method

Solution.

Mean (x)=a + ∑ fiyi/ N× h = 105 +2/30×30 = 107
Variance (σ²) = h²/N² [N∑ fiyi²-(∑ fiyi)²]
=(30)²/(30)² [30×76 - (2)²]
=2276

Q: How are variance and standard deviation related?

Q: What common parameters do both standard deviation and variance use?

Q: Are variance and standard deviation considered in the same unit of measurement?

Q: Why is the standard deviation always positive?

Q: Give an example of where standard deviation and variance are used?