How To Calculate Variance Of A Data Set

Statistics normally means taking the values of some quantities and by plotting information technology or past arranging it in a meaningful manner. It gives usa the procedure by which we tin can collect, analyze, translate, present and organize information. Variance is an important topic in statistics. Variance is a mensurate of central dispersion. It is the average of the squared divergence from the mean. Mean is the average of a given set of numbers. In this commodity, we volition talk over the steps to observe variance.

What is Variance?

The boilerplate of the squared differences from the mean is known as variance.

Variance is denoted past σ².

How to calculate variance stride by pace?

You can use the following steps to calculate variance.
Step ane. Calculate the mean of the given numbers.

Step 2. Then for each number, subtract the hateful and find the foursquare of the departure.

Pace 3. Calculate the average of those squared differences.

Formula

x₁, x_ii, …., x_nare the north observations.

Solved Examples

Example one: The means of five observations is 4 and their variance is five.ii. If three of these observations are i, 2 and 6, then the other two are

(A) 2 and nine

(B) three and eight

(D) 5 and 6

Solution:

Given no. of observations = five

Hateful = 4

Variance = five.2

Let unknown values be a and b.

[(1-iv)^two+ (two-four)² + (vi-4)^two + (a-four)² + (b-iv)^two ]/5= 5.2

(-three)^two + 2² + ii² + (a-iv)^two + (b-4)² = 26

nine + iv + 4 + (a-four)² + (b-4)² = 26

(a-4)² + (b-4)² = 26 – 17

(a-four)² + (b-four)² = 9

Substitute the given options in the higher up equation.

We observe that 4 and 7 satisfy the above equation.

Hence option C is the answer.

Example 2:

The variance of the starting time l even natural numbers is:

(A) 833/4

(B) 833

(D) 437/4

Solution:

The first 50 even natural numbers are ii, 4, 6,…100.

Variance is given by

= [(2^two+four²+…100²)/50 ] – [(2+4+6+…100)/50]ⁱⁱ

Nosotros volition separate this and solve it.

(2²+4²+…100ⁱⁱ)/l = ii²(1^two + 2²+3²+…l²)/50

Use the formula n(n+1)(2n+one)/6 to discover the sum of squares of natural numbers.

2ⁱⁱ(1² + 2²+3²+…50^two)/50 = (2ⁱⁱ/50)(fifty×51×101)/vi

= 3434

[(2+4+half-dozen+…100)/50]² = [2(1+two+three+..+fifty)/50]ⁱⁱ {apply the formula (due north/2)(2a+(n-i)d) to detect sum of n natural numbers.}

= [(2/50)(50/2)(2+49)]²

= 51²

= 2601

σ ²= 3434-2601 = 833

Hence option (B) is the respond.

Example three:

In an experiment with 15 observations on x, the following results were bachelor

\(\brainstorm{assortment}{l}\sum x^2 = 2830, \sum ten = 170\end{array} \)

On observation that was xx, was found to exist incorrect and was replaced past the right value thirty. And then the corrected variance is

(A) 78.00

(B) 188.66

(D) 8.33

Solution:

Given Σx = 170

Σx² = 2830

n = 15

20 was replaced by thirty. So change in Σx = 10

New Σx = 170+ten = 180

New Σxⁱⁱ = 2830+30²-20^two

= 2830 + 900 – 400

= 2830 + 500

= 3330

New variance = (3330/15) – (180/fifteen)²

= 222 -12²

= 222 – 144

= 78

Hence option (A) is the answer.

Oft Asked Questions

Give the equation for finding the variance.

Variance is given by σⁱⁱ = ∑(11-x̄)^two/N. x_i, x_two, …., ten_North are the Due north observations.

Define variance in statistics.

The average of the squared differences from the mean is chosen variance.

How practise you notice variance if standard divergence is given?

The square of the standard deviation gives the variance.

How practise you lot announce variance?

Variance is denoted by σ².