Statistics Formulas & Basic Concepts

Statistic formulas for class 8th, 9th, 10th, Calculation of mean, mode, median, mean deviation about mean & median, standard deviation, variance.

Statistics For Class 10

IMPORTANT INFORMATION

* For finding Mean class interval need not to be continuous.

* For finding Mode and Median class interval should be continuous.

* For making class interval continuous we should subtract 0.5 from the lower limits and add 0.5 to the upper limits of all class intervals if the difference of upper limit of first class interval and lower limit of second class interval is 1.

* For finding Mode and median, height of class interval need not to be equal or class interval may be irregular.

* If median class is the first class –interval then cumulative frequency (Cf) of the preceding class interval should be taken as zero.
* If Modal class is the first class –interval then f₀ of the preceding class interval should be taken as zero.
* If Modal class is the last class –interval then f₂ of the succeeding class interval should be taken as zero.

MEAN FOR UNGROUPED DATA

$\mathbf{Mean=\frac{Sum\;of\;all\;observations}{Number\;of\;observations}$

$\mathbf{Mean\left(\overline{X}\right)=\frac{\sum x_{i}}{n}$

If frequecy of the observations is also given then

$\mathbf{Mean\left(\overline{X}\right)=\frac{\sum f_{i}x_{i}}{\sum f_{i}}\;\;or\;\;\;\;\frac{\sum f_{i}x_{i}}{n}$

MEAN FOR A GROUPED DATA

There are three methods for finding the mean of grouped data.

1) DIRECT METHOD

$\mathbf{Mean\left(\overline{X}\right)=\frac{\sum f_{i}x_{i}}{\sum f_{i}}\;\;or\;\;\;\;\frac{\sum f_{i}x_{i}}{n}$

Where: x_i = mid-values

$\mathbf{x_{i}=\frac{Lower\:limit+Upper\:limit}{2}$

$\mathbf{\sum f_{i}=sum\;of\;all\;frequencies$

Table for finding Mean by Direct Method

C-I

x_i

f_i

f_ixi

$\mathbf{\sum f_{i}}$

$\mathbf{\sum f_{i}x_{i}}$

Range = Highest value - Lowest value
Class size = Upper limit - Lower limit = Height of the interval

2) ASSUMED MEAN METHOD

$\mathbf{Mean\left(\overline{X}\right)=a+\frac{\sum f_{i}d_{i}}{\sum f_{i}}\;\;or\;\;\;\;a+\frac{\sum f_{i}d_{i}}{n}}$

Where: a = assumed mean, d_i = x_i - a

Table for finding Mean by Assumed Mean Method

C-I

x_i

f_i

d_i = x_i - a

f_id_i

$\mathbf{\sum f_{i}}$

$\mathbf{\sum f_{i}d_{i}}$

3) STEP DEVIATION METHOD

$\mathbf{Mean\left(\overline{X}\right)=a+h\times\frac{\sum f_{i}u_{i}}{\sum f_{i}}\;\;or\;\;\;\;a+h\times\frac{\sum f_{i}u_{i}}{n}$

Where h is the height of the interval

$\mathbf{u_{i}=\frac{x_{i}-a}{h}$

Table for finding Mean by Step Deviation Method

C-I

x_i

f_i

$\mathbf{u_{i}=\frac{x_{i}-a}{h}$

$\mathbf{f_{i}u_{i}$

$\mathbf{\sum f_{i}}$

$\mathbf{\sum f_{i}u_{i}$

MODE

(FOR UNGROUPED DATA)

Most frequent observation is called mode, in other words, the observation whose frequency is maximum is called mode.

MODE :- (FOR GROUPED DATA)

First of all find the modal class

MODAL CLASS:- Class corresponding to the highest frequency is called the modal class.

$\mathbf{MODE=l+\left(\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}}\right)\times h$

Where:- l is the lower limit of the model class

f₁ = Frequency of the model class

f₀ = Frequency of the class preceeding the model class.

f₂ = Frequency of the class succeeding the model class.

MEDIAN

FOR UNGROUPED DATA

ALGORITHM:-

a) First write the given observation in ascending order.

b)Find the number of terms let it is n.

If n is odd then

$\mathbf{Median=\left(\frac{n+1}{2}\right)^{th}term$

If n is even then

$\mathbf{Median=\frac{1}{2}\left[\left(\frac{n}{2}\right)^{th}term\;+\left(\frac{n}{2}+1\right)^{th}term\right]$

MEDIAN FOR GROUPED DATA

ALGORITHM:-

1) Find the cumulative frequency of the observations.

$\mathbf{2)\;Find\;\;\sum f_{i}=N,\;then\;find\;\;\frac{N}{2}$

3) Then find the Median class and then Median.

Median Class = Class corresponding to Cf ≥ n/2 is called Median Class.

$\mathbf{Median=l+\left(\frac{\frac{N}{2}-c.f}{f}\right)\times h$

Where : l = Lower limit of Median Class

Cf = Commulative frequency of the class preceeding the Median Class.

f = Frequency of the Median Class.

h = Height of the Median Class.

Table for finding the Median

C-I

f_i

$\mathbf{\sum f_{i}}$

EMPIRICAL FORMULA

It is the formula which shows the relationship between mean, mode and median

Mode = 3 Median - 2 Mean

Note: Empirical formula does not provide the same result in different situations. So, students should use it only when asked otherwise avoid it.

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Statistics for Class 10+1

Measure of central tendency:

Representative value of the given data is called the central tendency. Meam, Mode, Median are the three measures of central tendency. A measure of central tendency gives us a rough idea where the data points are centered.

Dispersion

Measures of central tendency are not sufficient to give complete information about a given data. Variability is another factor which is required to be studied under statistics. Like measures of central tendency we want to have a single number to describe variability. This single number is called Dispersion.

Measures of Dispersion:

There are following measures of Dispersion:

i) Range, ii) Quartile Deviation, iii) Mean Deviation iv) Standard Deviation

In this chapter we will study all measures except Quartile Deviation.

Range : Maximum Value - Minimum value

Mean Deviation:

$\mathbf{Mean\:of\:Deviations=\frac{Sum\:of\:deviations}{Number\:of\:Deviations}}$

In this chapter we discuss two types of Mean Deviation.

i) Mean Deviation about mean ii) Mean Deviation about Median

MEAN DEVIATION ABOUT MEAN

FOR UNGROUPED DATA

$\mathbf{Mean\;Deviation\;About(\overline{X})=\frac{\sum\left|x_{i}-\overline{x}\right|}{n}$

FOR GROUPED DATA

$\mathbf{Mean\;Deviation\;About(\overline{X})=\frac{\sum f_{i}\left|x_{i}-\overline{x}\right|}{n}$

C-I

x_i

f_i

f_ixi

$\mathbf{\left|x_{i}-\bar{x}\right|}$

$\mathbf{f_{i}\left|x_{i}-\bar{x}\right|$

$\mathbf{\sum f_{i}}$

$\mathbf{\sum f_{i}x_{i}}$

$\mathbf{\sum f_{i}\left|x_{i}-\bar{x}\right|$

MEAN DEVIATION ABOUT MEDIAN

FOR UNGROUPED DATA

$\mathbf{M.D.(M)=\frac{\sum\left|x_{i}-M\right|}{N}$

Where M = Median of the data.

FOR GROUPED FREQUENCY

$\mathbf{M.D.(M)=\frac{\sum f_{i}\left|x_{i}-M\right|}{N},$

Where N is the sum of all frequencies.

C-I

x_i

f_i

$\mathbf{\left|x_{i}-M\right|}$

$\mathbf{f_{i}\left|x_{i}-M\right|$

$\mathbf{\sum f_{i}}$

$\mathbf{\sum f_{i}\left|x_{i}-M\right|$

Note: For calculating mean, class interval may or may not be continuous. But for calculating median, class intervals should be continuous.

Limitations of Mean Deviation about Mean and about Median

While calculating Mean deviation about mean and about median we take absolute values and ignore the negative sign. So this calculation is not become very scientific. Also in many cases it gives unsatisfactory results.

This imply that we need another measure of Dispersion. Standard Deviation is such a measure of central tendency.

While calculating mean deviation about mean or median, the absolute values of the deviations were taken otherwise deviations may cancel among themselves.

Another way to overcome this difficulty which arose due to the sign of deviations, is to take square of all the deviations. Mean of squares of deviations about mean is called Variance.

Variance and Standard Deviation.

VARIANCE

Mean of the squares of the deviations from the mean is called the variance.

VARIANCE FOR UNGROUPED DATA

$\mathbf{Variance(\sigma^{2})=\frac{\sum\left(x_{i}-\overline{x}\right)^{2}}{N}}$

Where n is the sum of all frequencies

VARIANCE FOR GROUPED DATA

$\mathbf{Variance(\sigma^{2})=\frac{\sum f_{i}\left(x_{i}-\overline{x}\right)^{2}}{N},Where\:N=\sum f_{i}}$

Important Result

$\mathbf{Derivation\:of\:\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}-\left(\bar{x}\right)^{2}}$

$\mathbf{\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}+\frac{\sum(\bar{x})^{2}}{N}-\frac{\sum 2x_{i}\bar{x}}{N}}$

$\mathbf{\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}+\frac{(\bar{x})^{2}N}{N}-2\bar{x}\left(\frac{\sum x_{i}}{N}\right)}$

$\mathbf{\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}+(\bar{x})^{2}-2\bar{x}\bar{x}}$

$\mathbf{\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}+(\bar{x})^{2}-2(\bar{x})^{2}}$

$\mathbf{\frac{1}{N}\sum\left(x_{i}-\bar{x}\right)^{2}=\frac{\sum(x_{i})^{2}}{N}-(\bar{x})^{2}}$

NCERT Exercise 15.2
Q 2: Find mean and variance of first n natural numbers?

Ans: Let first n natural numbers are 1, 2, 3, 4, ................, N

$\mathbf{Mean(\bar{X})=\frac{1+2+3....n}{n}=\frac{n(n+1)}{2n}=\frac{n+1}{2}}$

$\mathbf{Variance(\sigma^{2})=\frac{1}{n}\sum(x_{i}-\bar{x})^{2}}$

$\mathbf{=\frac{\sum\left(x_{i}\right)^{2}}{n}-(\bar{x})^{2}=\frac{n(n+1)(2n+1)}{6n}-\left(\frac{n+1}{2}\right)^{2}}$

$\mathbf{=\left(\frac{n+1}{2}\right)\left[\frac{2n+1}{3}-\frac{n+1}{2}\right]=\left(\frac{n+1}{2}\right)\left(\frac{n-1}{6}\right)}$

$\mathbf{=\frac{n^{2}-1}{12}}$