Class 10 Maths notes for Statistics

Table of content

• Flash back from IX Class Statistics
• Mean for Ungroup Frequency table
• Mean for group Frequency table
• Various ways to calculate mean
• Mode for grouped frequency table
• Cumulative Frequency chart
• Median of a grouped data frequency table
• Empirical Formula between Mode, Mean and Median

Mean for Ungroup Frequency table

Here is the ungroup Frequency table

Mean is given by

Greek letter ∑ (capital sigma) means summation
Example
A survey was conducted by a group of students as a part of their environment awareness programmes, in which they collected the following data regarding the number of plants in 30 houses in a locality. Find the mean number of plants per house.

 
Solution



Mean = 172/30 = 5.73
 

Mean for group Frequency table

In these distribution, it is assumed that frequency of each class interval is centered around its mid-point i.e class marks

Mean can be calculated using three method

(a) Direct method

This method can be very calculation intensive if the values of f and x are large.We have big calculations and chance of making mistake is quite high

Steps involved in finding the mean using Direct Method
(1) Prepare a frequency table with the help of class marks
(2) Multiply f_i x _i and find the sum of it.
(4) Use the above formula and find the mean.
Example The following table shows the weights of 10 children: 



Find the mean by using direct method.
Solution: 


 
So, Mean would be
=698/10  = 69.8 kg

(b) Assumed mean method

Where
a=> Assumed mean
d_i   => x_i –a
This method is quite useful when the values of f and x are large. It makes the calculation easiar.In this method we take some assumed mean and calculate the deviation from it and then calculate mean using above formula
Steps involved in finding the mean using Assumed Mean Method
(1) Prepare a frequency table.
(2) Choose A and take deviations d_i = x_i - A of the values of x_i .
(3) Multiply f_i d _i and find the sum of it.
(4) Use the above formula and find the mean.
Example Example
The following table shows the weights of 10 children: 


Find the mean by using Assumed Mean method.
Solution: 
Let the assumed mean = A = 71


 
So, Mean would be
=71-12/10  = 69.8 kg

c) Step deviation Method

Where
a=> Assumed mean
u_i   => (x_i –a)/h
This method is quite useful when the values of f and x are large. It makes the calculation further easiar by dividing the deviation from common factor.
Steps involved in finding the mean using Step Deviation Method
(1) Prepare a frequency table.
(2) Choose A and h and take u_i = (x_i –a)/h of the values of x_i .
(3) Multiply f_i u _i and find the sum of it.
(4) Use the above formula and find the mean.
Example
The following table shows the weights of 10 children: 

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Find the mean by using Step Deviation method.
Solution: 
Let the assumed mean = A = 71  and h=2


So, Mean would be
=71+ (-6/10) 2  = 69.8 kg
 

Mode for grouped frequency table

Modal class: The class interval having highest frequency is called the modal class and Mode is obtained using the modal class
Mode formula is given as

Where
l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f₁ = frequency of the modal class,
f₀ = frequency of the class preceding the modal class,
f₂ = frequency of the class succeeding the modal class.
Example
The following table shows the ages of the patients admitted in a hospital during a year



Find the mode
Solution
Modal class = 35 – 45, l = 35, class width (h) = 10, f₁ = 23, f₀ = 21 and f₂ = 14
Substituting the values in the Mode formula given above we get

Mode= 36.8 year

Cumulative Frequency chart

The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given class.

Cumulative Frequency chart will be like

The above table cumulative frequency distribution of the less than type. We can similary make it like below  

 
The table above is called a cumulative frequency distribution of the more than type.

Median of a grouped data frequency table

How to find Median of a grouped data frequency table
(1) For the given data, we need to have class interval, frequency distribution and cumulative frequency distribution
(2)Then we need to find the median class
How to find the median class
(a) we find the cumulative frequencies of all the classes and n/2
(b)We now locate the class whose cumulative frequency is greater than (and nearest to) n/2
(c)That class is called the median class
(3) Median is calculated as per the below formula

Where
l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal)
Example A survey regarding the heights (in cm) of 60 girls  of a school was conducted and the following data was obtained:

Height (in cm)	Number of girls
Less than 140	4
Less than 145	11
Less than 150	29
Less than 155	40
Less than 160	46
Less than 165	60

Find the median height
Solution
To calculate the median height, we need to find the class intervals and their corresponding frequencies.
The given distribution being of the less than type, 140, 145, 150, . . ., 165 given the upper limits of the corresponding class intervals. So, the classes should be below 140, 140 - 145, 145 - 150, . . ., 160 - 165. Observe that from the given distribution, we find that there are 4 girls with height less than 140, i.e., the frequency of class interval below 140 is 4 . Now, there are 11 girls with heights less than 145 and 4 girls with height less than 140. Therefore, the number of girls with height in the interval 140-145 will be 11-4=7. Similarly, other can be calculated
 

Class interval	Frequency	Cumulative Frequency
Below 140	4	4
140-145	7	11
145-150	18	29
150-155	11	40
155- 160	6	46
160- 165	14	60

 
So, n =60 and n/2=30 And cumulative frequency which is greater than and nearest to 30 is 40 , So median class 150-155
l (the lower limit) = 150,
cf (the cumulative frequency of the class preceding 150 - 155) = 29,
f (the frequency of the median class 150 - 151) = 11,
h (the class size) = 5.
Now by Median Formula

= 150 + [(30-29)/11]5
=150.45 cm

Empirical Formula between Mode, Mean and Median

Empirical Formula between Mode, Mean and Median is given as 3 Median=Mode +2 Mean

Graphical representation of Cummulative frequency distribution

We can represent Cummulative frequency distribution on the graph also. To represent the data in the table graphically, we mark the upper limits of the class intervals on the horizontal axis (x-axis) and their corresponding cumulative frequencies on the vertical axis (y-axis), choosing a convenient scale.
When we draw the graph for the cumulative frequency distribution of the less than type.The curve we get is called a cumulative frequency curve, or an ogive (of the less than type).

When we draw the graph for the cumulative frequency distribution of the more than type.The curve we get is called a cumulative frequency curve, or an ogive (of the more than type).

When we plot both these curve on the same axis, The two ogives willintersect each other at a point. From this point, if we draw a perpendicular on the x-axis, the point at which it cuts the x-axis gives us the median

Summary of Statistics Class 10 Formulas

Also Read

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• Notes
  • Statistics Class 10 notes
• Assignments
  • Statistics Class 10 Problem and Solutions
• NCERT Solutions
  • NCERT Solutions Statistics Class 10 Ex 14.1
  • NCERT Solutions Statistics Class 10 Ex 14.2

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