Statistics Class 11 Notes Mathematics Chapter 15 - CBSE

Chapter : 15

What Are Statistics ?

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Introduction

It is the discipline that concerns the collection, organisation, analysis, interpretation and presentation of data.

In earlier classes, we studied about the methods of presenting the data collected in the graphical from and tabular form. There are three measures of central tendency :

(i) Arithmetic mean (ii) Median (iii) Mode These measures gives a rough idea about the points where data are centered.

In this chapter we will study about mean deviation and standard deviation, in order to have a better idea as to know how the data are scattered.

Range

It is a difference between two extreme observations of the distribution

Range = Largest observation – Smallest observation

Mean Of Ungrouped Data

The ungrouped data is a set of data not sorted into categories or classified. It is basically a list of numbers.

Direct method

Mean of ungrouped data is calculated by :

$$\bar x=\frac{x_{1} + x_{2}+ ...+ x_{n}}{n}$$

Deviation method

In this method, we assume a mean a. Then, compute d_i = x_i – a, i.e. deviation of the values of data from assumed mean. Then mean of given data is

$$\bar x = a + \frac{\Sigma d_{i}}{n}$$

Mean Of Discrete Distribution

Discrete frequency distribution consist of 'm' distinct values x₁ , x₂ , ... x_m with frequencies f₁ , f₂ , ...fm respectively. There are two ways of calculating mean :

Direct method

The mean of the given data is

$$\bar{x} =\frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}$$

Deviation method

In this method, we assume a mean suppose 'a'.

Then calculate deviation i.e., d_i = x_i – a

Then, mean of the given data is :

$$\bar x = a + \frac{\Sigma f_{i}d_{i}}{\Sigma f_{i}}$$

Deviation method

In this method, we assume a mean suppose 'a'.

Then calculate deviation i.e., d_i = x_i – a

Then, mean of the given data is :

$$\bar x = a + \frac{\Sigma f_{i}d_{i}}{\Sigma f_{i}}$$

Mean Of Continuous Distribution

For computing the mean of continuous frequency distribution, with correspoding frequencies di , we first compute the midpoint of the class intervals and denote them by x_i.

There are three method for calculating mean of the continuous distribution.

Direct method

The mean of given data is :

$$\bar x = \frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}$$

Deviation method

In this method, take an assumed mean 'a' and class with 'h'.

$$\text{Compute U}_{i} =\frac{x_{i}-a}{h}\space\text{i.e.,}$$

the step-deviations of the mid-points of class intervals from assumed mean.

Then, mean is given by.

$$\bar{x} = a + h\frac{\Sigma f_{i}U_{i}}{\Sigma f_{i}}$$

Median Of Ungrouped Data

For finding the median of ungrouped data, first arrange it in ascending order.

Now, calculate the median using the following formula :

$$\text{Median} =\bigg(\frac{n+1}{2}\bigg)^{\text{th}}\text{term,}\\\text{if n is odd}\\\text{Median = }\\\frac{\bigg(\frac{n}{2}\bigg)^{\text{th}}\text{term} + \bigg(\frac{n}{2}+1\bigg)^{\text{th}}\text{term}}{2},\\\text{if n is even}$$

Median Of Discrete Distribution

Consider a discrete frequency distribution having 'm' distinct values x₁ , x₂... x_n with frequencies f₁ , f₂ ... f_n respectively.

Then, arrange the data in (values of x_i ) in ascending order. Find cumulative frequency.

Denote n = ± ∑f_i .

Then, we calculate median using the formula,

$$\text{Median =}\bigg(\frac{n+1}{2}\bigg)^{\text{th}}\space\text{term},\\\text{if n is odd}\\\text{Median = }\\\frac{\bigg(\frac{n}{2}\bigg)^{\text{th}} +\bigg(\frac{n}{2}+1\bigg)^{\text{th}}\text{term}}{2},\\\text{if n is even}$$

Median Of Continuous Frequency Distribution

For finding the median, we first find the mid-points of the class intervals and denote them by x_i.

Then, calculate the cumulative frequency.

Denote n = ∑f_i Identify the median class i.e.,

the class containing the cumulative frequency

$$\frac{x}{2}.$$

$$\text{Then, Median =}\\ l +\bigg(\frac{\frac{n}{2} - c}{f}\bigg)× h$$

where,

l = lower limit of median class

h = width limit of median class

f = frequency of median class

c = cumulative frequency just preceding median class.

Mean Deviation

A number that represents the mean of the deviations from that number.

Mean deviation about mean

The AM of the numerical deviations of the observations from the mean of the data is called mean deviation about mean.

Mean deviations about median

The AM of the numerical deviations of the observations from the median of the data is called the mean deviation about median.

Mean deviation for ungrouped data: Consider the ungrouped data as x₁ , x₂ , .... x_n consisting of 'n' values. Let x be its mean and M be its median. Then, median deviation about mean

$$(\bar{x}) =\sum^{n}_{i=1}\frac{|x_{i}-\bar{x}|}{n}\\\text{where}\space\bar{x}\space\text{is the mean}$$

Mean deviation about median

$$(\text{M}) =\sum^{n}_{i=1}\frac{|x_{i}-\text{M}|}{n}$$

where M is the median

Mean deviation for discrete distribution: Consider the data as x₁, x₂ , .... x_n having frequencies f₁ , f₂... f_nrespectively.

Then, mean deviation about mean is :

$$\text{MD}(\bar{x}) =\sum^{n}_{i=1}\frac{f_{i}|x_{i}-\bar{xx}|}{\displaystyle\sum_{i=1}^nf_{i}}\\=\displaystyle\sum_{i=1}^n\frac{f_{i}|x_{i} -\bar{x}|}{\text{N}}\\\text{where,}\space x =\frac{\displaystyle\sum^{n}_{i=1}f_{i}x_{i}}{\displaystyle\sum f_{i}}$$

Mean deviation about median is :

$$\text{MD(M)} =\displaystyle\sum^{n}_{\text{i = 1}}\frac{f_{i}|x_{i} - \text{M}|}{\text{N}}$$

where, N is median of the data and N = ∑f_i.

Variance And Standard Deviation

Variance

Mean of squares of the deviations from the mean is called the variance and is denoted by σ² .

Steps followed for calculating

Calculate the mean of the given data, denoted by

$$\bar{x}.\space \text{Then, the variance for the data}\\x_{1},x_{2},.....x_{n}\space\text{is given by}$$

$$\sigma^{2} =\frac{1}{n}\displaystyle\sum_{i=1}^n(x_{i} -\bar{x})^{2}$$

Variance for the grouped data (discrete or continuous) is :

$$\sigma^{2} =\frac{\Sigma f_{i}(x_{i} -\bar{x})^{2}}{\Sigma f_{i}}$$

Standard Deviation

The positive square root of variance is called standard deviation and it is denoted by σ. For ungrouped data,

$$\text{Standard deviation,}\\\space\sigma =\sqrt{\frac{\Sigma (x_{i}-\bar{x})^{2}}{n}}$$

For grouped (discrete/continuous) data,

$$\text{Standard deviation,}\\\sigma =\frac{\sqrt{\Sigma f_{i}(x_{i}-\bar{x}_{i})^{2}}}{n}$$

Short-cut method for calculating S.D.

For ungrouped data,

$$\text{Variance,}\space\sigma^{2}=\\\frac{1}{n}\begin{Bmatrix}\displaystyle\sum_{i=1}^n (x_{i} -\bar{x})^{2}\end{Bmatrix}\\=\frac{1}{n}\begin{Bmatrix}\displaystyle\sum^{n}_{i=1}(x_{i}^{2} - 2x_{i}\bar{x} + x^{2})\end{Bmatrix}\\=\frac{\Sigma x_{i}^{2}}{n}- 2\bar{x}\bigg(\frac{\Sigma x_{i}}{n}\bigg) + \frac{n\bar{x}^{2}}{n}\\=\frac{\Sigma x_{i}^{2}}{n}-2\bar{x}.\bar{x} +x^{2}\\\therefore\sigma^{2} =\frac{\Sigma_{i}^{2}}{n}-\bar{x}^{2}$$

Standard deviation,

$$\sigma^{2} =\sqrt{\frac{\Sigma x_{i}^{2}}{n} - \bar{x}^{2}}$$

For grouped data,

$$\text{Variance,}\space\sigma^{2}=\frac{\Sigma f_{i}(x_{i}-\bar{x})^{2}}{\Sigma f_{i}}\\=\frac{\Sigma f_{i}(\bar{x_{i}}^{2} - \bar{x}^{2} - 2x_{i}\bar{x})}{\Sigma f_{i}}\\=\frac{\Sigma f_{i}x_{i}^{2}}{\Sigma f_{i}} + \frac{\Sigma\bar{x}^{2}f_{i}}{\Sigma f_{i}} + \frac{\Sigma 2 x_{i}\bar{x}}{\Sigma f_{i}}\\=\frac{\Sigma f_{i}x_{i}^{2}}{\Sigma f_{i}} + \bar{x}^{2} -\bar{2x}^{2}\\\therefore\space \sigma^{2}=\frac{\Sigma f_{i}x_{i}^{2}}{\Sigma f_{i}}-\bar{x}^{2}\\\text{or}\space \sigma^{2} =\frac{\Sigma f_{i}x_{i}^{2}}{\Sigma f_{i}}-\bigg(\frac{\Sigma f_{i}f_{i}}{\Sigma f_{i}}\bigg)^{2}$$

and Standard deviation,

$$\sigma =\sqrt{\frac{\Sigma f_{i}x_{i}^{2}}{\Sigma f_{i}}-\bigg(\frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}\bigg)^{2}}$$

Note: Coefficient of variation is given by

$$\text{C.V.}=\frac{\sigma}{\bar{x}}×100$$