Here, you will learn learn formula for variance and standard deviation and relationship between variance and standard deviation.
Let’s begin –
Variance and Standard Deviation :
The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by σ2 or var(x).
The positive square root of the variance are called the standard deviation. It is denoted by σ or S.D.
Hence standard deviation = + √variance
Formula for Variance :
(i) for ungrouped distribution :
σ2x = ∑(xi–ˉx)2n
σ2x = ∑xi2n – ˉx2
= ∑xi2n – (∑xin)2
σ2d = ∑di2n – (∑din)2, where di = xi – a
(ii) for frequency distribution :
σ2x = ∑fi(xi–ˉx)2N
σ2x = ∑fixi2N – ˉx2
= ∑fixi2N – (∑fixiN)2
σ2d = ∑fidi2n – (∑fidin)2, where di = xi – a
σ2d = h2[∑fiui2n – (∑fiuin)2], where ui = xih
(iii) Coefficient of Standard Deviation = σˉx
Coefficient of variation = σˉx × 100 (in percentage)
Example : Find the variance and standard deviation of first n natural numbers.
Solution : We know that,
σ2x = ∑xi2n – (∑xin)2
= ∑n2n – (∑nn)2 = n(n+1)(2n+1)6n – [n(n+1)2n]2 = n2–112
Standard Deviation = √variance = √n2–112
Example : Find the Coefficient of variation in percentage of first n natural numbers.
Solution : We know that
Mean ˉx = n+12,
Variance = σ2x = n2–112
Standard Deviation σ = √variance = √n2–112
Coefficient of variation = σˉx × 100
= √n2–112 × (2n+1) × 100
= √(n–1)3(n+1) × 100
Hope you learnt what is the formula for variance and standard deviation, learn more concepts of statistics and practice more questions to get ahead in the competition. Good luck!