Variance: The variance is the measure of how much spread or variability is present in the sample.

If the all the numbers in the sample are very close to each other, the variance is close to zero.

If the numbers well dispersed or scatter the variance will tend to be large.

The variance of a set of data is defined as the sum of squares of the deviations of observations from their mean divided by their number of observations.

It is denoted by σ2 (sigma square).

Mathematically, For Ungrouped data variance can be written as

That is, the mean of squared deviations of observations from their mean is known as the variance.

Then find variance of the given data 5,9,11, 4,3, 20

Variance of Group data can be mathematically written as

Standard Deviation: The positive square root of variance is known as Standard deviation.

Standard deviation is denoted by σ (sigma) or s

Standard Deviation is written as

The standard deviation measures the absolute variation of the distribution. The greater the amount of variation, the greater the standard deviation (SD).

A small standard deviation means a high degree of uniformity and homogeneity of the observations. A large standard deviation means the opposite.

1. Find the standard deviation from the weekly wages of the workers working in a factory

Worker Wages Worker WagesA 320 F 340B 310 G 345C 315 H 330D 325 I 333E 350 J 342

An analysis of production rejects result in the following table. Find standard deviation

No. of rejected No. Of operatorProduct 20-25 525-30 1530-35 2835-40 4240-45 1545-50 1250-55 3

An analysis of production rejects result in the following table. Find standard deviation

class x f fx fx^220-25 22.5 5 113 2531.325-30 27.5 15 413 1134430-35 32.5 28 910 2957535-40 37.5 42 1575 5906340-45 42.5 15 638 2709445-50 47.5 12 570 2707550-55 52.5 3 158 8268.8TOTAL 120 4375 164950

mean 36.5 1374.6

Mathematical Properties of Sd:

1. The Standard Deviation(SD) of first n natural numbers can be obtain as

Hence find the standard deviation of the data 1,2,3,4………………………………..500

2. Standard deviation is independent of change of origin but dependent of scale.

3. For symmetrical distributionMean±1σ covers 68.27% data

Mean±2σ covers 95.45% dataMean±3σ covers 99.73% data4. Relation between measures of variations

Q.D=2/3 SDA.D=4/5 SD

Merits of SD:

1. SD is the best measures of variation2. SD is used in comparing two or

more distribution.3. It is possible to calculate the

combined SD4. SD is most prominently used for

further statistical work.

Limitation of SD: 1.It is difficult to compute.2.It gives more weight to

extreme values and less weight to near mean.

Example: The performance of run of two player is given. Select who is better one of the two?

Tamim Mushfik

No. of Matches 100 100Average of Run 120 80SD of Run 45 20

Co-efficient of Variation(CV): When we do not get an accurate picture just by comparing two sets of data by standard deviation then Co-efficient of variation solve this difficulty.

Co-efficient of variation(CV) is computed as a ratio of the standard deviation to the mean of the same distribution.

Mathematically, Co-efficient of Variation(CV)

# For which sample Co-efficient of variation is greater is said to be less consistent, less uniform, less stable or less homogeneous and more variability.

# For which Co-efficient of variation is less is said to be more consistent, more uniform, more stable or more homogeneous and less variability.

The price of tea company shares in Dhaka and Chittagong share markets during the last Four months are recorded as below:

Month Dhaka ChittagongJanuary 106 108February 120 112March 100 110April 130 114In Which Market Share prices are more

Stable?

Home Work: A purchasing agent obtained samples of 60 watt bulbs from two companies. He had the samples tested in his own laboratory for length of life with the following results:

Length of life Company A Company B(in Hours) 1700-1900 10 31900-2100 16 402100-2300 20 122300-2500 8 32500-2700 6 2

i) Which company’s bulb do you think are better in terms average life?

ii) If prices of bulbs are the same, which company’s bulbs would you buy and why?

For Ungrouped data variance can

For Ungrouped data variance

Standard Deviation is written as

The Standard Deviation(SD) of first n natural numbers

Co-efficient of Variation(CV)