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Statistics Class 10 Notes Maths Chapter 13

Mean

The mean value of a variable is defined as the sum of all the values of the variable divided by the number of values.

Statistics Class 10 Notes Maths Chapter 13

Median

The median of a set of data values is the middle value of the data set when it has been arranged in ascending order.  That is, from the smallest value to the highest value
Median is calculated as:

Statistics Class 10 Notes Maths Chapter 13

Where n is the number of values in the data.
If the number of values in the data set is even, then the median is the average of the two middle-value.

Mode

Mode of statistical data is the value of that variable that has the maximum frequency.

Mean for Ungroup Frequency Table

Here is the ungroup frequency table:

Statistics Class 10 Notes Maths Chapter 13Mean is given by:
Statistics Class 10 Notes Maths Chapter 13

Greek letter ∑ (capital sigma) means summation.

Mean for Group Frequency Table

Statistics Class 10 Notes Maths Chapter 13In this distribution, it is assumed that the frequency of each class interval is centered around its mid-point i.e class marks.
Statistics Class 10 Notes Maths Chapter 13

Mean can be calculated using three methods:

(a) Direct Method
Statistics Class 10 Notes Maths Chapter 13This 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

  • Prepare a frequency table with the help of class marks
  • Multiply fand  xand find the sum of it.
  • Use the above formula and find the mean.

Example: The following table shows the weights of 10 children: 
Statistics Class 10 Notes Maths Chapter 13

Find the mean by using the direct method.
Sol: 
Statistics Class 10 Notes Maths Chapter 13So, Mean would be
=698/10  = 69.8 kg.

(b) Assumed Mean Method
Statistics Class 10 Notes Maths Chapter 13

Where
a= Assumed Mean
di = xi –a

This method is quite useful when the values of f and x are large. It makes the calculation easier. In this method, we take some assumed mean calculate the deviation from it and then calculate the mean using the above formula.

Question for Chapter Notes: Statistics
Try yourself:
What is the mean value of a variable?
View Solution

Steps involved in finding the mean using the Assumed Mean Method

  • Prepare a frequency table.
  • Choose A and take deviations ui = (xi -a)/h of the values of xi .
  • Multiply fi ui and find the sum of it.
  • Use the above formula and find the mean.

Example: The following table shows the weights of 10 children:

Statistics Class 10 Notes Maths Chapter 13

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

Statistics Class 10 Notes Maths Chapter 13

So, Mean would be
= 71-12/10  = 69.8 kg
c) Step Deviation Method
Statistics Class 10 Notes Maths Chapter 13

Where
a= Assumed mean
ui  = (xi –a)/h
This method is quite useful when the values of f and x are large. It makes the calculation further easier by dividing the deviation from the common factor.

Steps involved in finding the mean using Step Deviation Method

  • Prepare a frequency table.
  • Choose A and h and take ui = (xi –a)/h of the values of xi .
  • Multiply fi u i and find the sum of it.
  • Use the above formula and find the mean.

Example: The following table shows the weights of 10 children: 

Statistics Class 10 Notes Maths Chapter 13

Find the mean by using Step Deviation method.
Sol:

Let the assumed mean = A = 71  and h=2
Statistics Class 10 Notes Maths Chapter 13

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
Statistics Class 10 Notes Maths Chapter 13

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

Statistics Class 10 Notes Maths Chapter 13Find the mode.
Sol:
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
Statistics Class 10 Notes Maths Chapter 13

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.
Statistics Class 10 Notes Maths Chapter 13Cumulative Frequency chart will be like

Statistics Class 10 Notes Maths Chapter 13

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

Statistics Class 10 Notes Maths Chapter 13

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

  • For the given data, we need to have class interval, frequency distribution and cumulative frequency distribution
  • 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
  • Median is calculated as per the below formula

Statistics Class 10 Notes Maths Chapter 13

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:

Statistics Class 10 Notes Maths Chapter 13

Find the median height.

Sol:
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

Statistics Class 10 Notes Maths Chapter 13

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

Statistics Class 10 Notes Maths Chapter 13

= 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

Question for Chapter Notes: Statistics
Try yourself:
What is the formula used to find the mean using the Assumed Mean Method?
View Solution

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 will intersect 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

Statistics Class 10 Notes Maths Chapter 13

The document Statistics Class 10 Notes Maths Chapter 13 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Statistics Class 10 Notes Maths Chapter 13

1. What is the difference between mean, median, and mode in statistics?
Ans. In statistics, the mean is the average of a set of numbers, calculated by adding up all the numbers and dividing by the total count. The median is the middle value in a set of numbers when they are arranged in numerical order. The mode is the number that appears most frequently in a set of numbers.
2. How can a cumulative frequency chart be used in statistics?
Ans. A cumulative frequency chart is used to show the total frequency of values up to a certain point in a dataset. It helps in understanding the distribution of values and can be used to find the median and quartiles of the data.
3. Is there a relationship between the mode, mean, and median in a dataset?
Ans. Yes, there is a relationship between the mode, mean, and median in a dataset. In a symmetrical distribution, the mode, mean, and median will be approximately equal. In a skewed distribution, the mode will be the highest point, the mean will be affected by the skewness, and the median will be the middle value.
4. How can the empirical formula be used to relate the mode, mean, and median in a dataset?
Ans. The empirical formula states that for a symmetrical distribution, the mode is approximately equal to 3 times the median minus 2 times the mean. This formula can be used to estimate the mode when the mean and median are known.
5. What is the significance of understanding the mean, median, and mode in statistical analysis?
Ans. Understanding the mean, median, and mode is crucial in statistical analysis as they provide insights into the central tendency and distribution of data. They help in summarizing and interpreting data, making comparisons between different datasets, and making informed decisions based on the characteristics of the data.
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