NCERT Class 10 Mathematics Statistics CBSE Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2022)

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Flash back from IX Class

A systematic record of facts or different values of a quantity is called data.

Features of the data

  • Statistics deals with collection, presentation, analysis, and interpretation of numerical data.
  • Arranging data in an order to study their salient features is called presentation of data.
  • Data arranged in ascending or descending order is called arrayed data or an array
  • Range of the data is the difference between the maximum and the minimum values of the observations
  • Table that shows the frequency of different values in the given data is called a frequency distribution table
  • A frequency distribution table that shows the frequency of each individual value in the given data is called an ungrouped frequency distribution table.
  • A table that shows the frequency of groups of values in the given data is called a grouped frequency distribution table
  • The groupings used to group the values in given data are called classes or class-intervals. The number of values that each class contains is called the class size or class width. The lower value in a class is called the lower class limit. The higher value in a class is called the upper class limit.
  • Class mark of a class is the mid-value of the two limits of that class.
  • A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution.
  • A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution

Bar Graph:

A bar graph is a pictorial representation of data in which rectangular bars of uniform width are drawn with equal spacing between them on one axis, usually the x-axis. The value of the variable is shown on the other axis that is the y-axis.

Histogram:

A histogram is a set of adjacent rectangles whose areas are proportional to the frequencies of a given continuous frequency distribution

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.

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

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 a statistical data is the value of that variable which has the maximum frequency

Mean for Ungroup Frequency table

Mean for Ungroup Frequency Table
Mark obtained 25354565
No of student 4102334

Mean is given by

Mean for group Frequency table

Mean for Group Frequency Table
Mark obtained 10 - 2525 - 4545 - 6565 - 85
No of student 4102334

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

Class mark =

Mean can be calculated using three method

a) Direct method

b) Assumed mean method

Where

a Assumed mean

c) Step deviation Method

Where

a Assumed mean

Mode for grouped frequency ta1e

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

Where

l = lower limit of the modal class,

h = size of the class interval (assuming all class sizes to be equal) ,

= frequency of the modal class,

= frequency of the class preceding the modal class,

= frequency of the class succeeding the modal class.

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
Class interval (Age)No of Insurance policies
15 - 202
20 - 254
25 - 3016
30 - 3520
35 - 4020
40 - 4512

Cumulative Frequency chart will be like

Cumulative Frequency Chart Will be Like
Age in yearsCumulative No of Insurance policies
Less than 20 years2
Less than 25 years6
Less than 30 years22
Less than 35 years42
Less than 40 years62
Less than 45 years74

Median of a grouped data frequency table

For the given data, we need to have class interval, frequency distribution, and cumulative frequency distribution

Median is calculated as

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)

Empirical Formula between Mode, Mean, and Median

3 Median = Mode + 2 Mean