Statistics Class 10 Notes: Chapter 14

CBSE Class 10 Maths Statistics Notes:-Download PDF Here

The brief notes on statistics for class 10 are given here. In this, we are going to discuss the important statistical concepts, such as grouped data, ungrouped data and the measures of central tendencies like mean, median and mode, methods to find the mean, median and mode, the relationship between them with more examples.

Introduction to Statistics

Ungrouped Data

Ungrouped data is data in its original or raw form. The observations are not classified into groups.

For example, the ages of everyone present in a classroom of kindergarten kids with the teacher is as follows:

3, 3, 4, 3, 5, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 27.

This data shows that there is one adult present in this class and that is the teacher.  Ungrouped data is easy to work with when the data set is small.

Grouped Data

In grouped data, observations are organized in groups.

For example, a class of students got different marks in a school exam. The data is tabulated as follows:

This shows how many students got the particular mark range. Grouped data is easier to work with when a large amount of data is present.

Frequency

Frequency is the number of times a particular observation occurs in data.

Class Interval

Data can be grouped into class intervals such that all observations in that range belong to that class.

Class width = upper class limit – lower class limit

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Mean

Finding the mean for Grouped Data when class Intervals are not given

For grouped data without class intervals,

Mean =

Finding the mean for Grouped Data when class Intervals are given

For grouped data with class intervals,

Mean =

Where $f_i$ is the frequency of $i^{th}$ class whose class mark is $x_i$.
Classmark =(Upper Class Limit+ Lower Class Limit)/2

Direct method of finding mean

Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Tabulate the product of the class mark and its corresponding frequency for each class. Calculate their sum $(\sum x_i{f}_i).$

Step 4: Divide the above sum by the sum of frequencies $(\sum f_i)$ to get the mean.

Assumed mean method of finding mean

Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Take one of the $x_i$’s (usually one in the middle) as the assumed mean and denote it by ${}^{}{a}^{\prime }$.

Step 4: Find the deviation of ${}^{}{a}^{\prime }$ from each of the $x_i^{\prime }s$
$d_i=x_i-a$

Step 5: Find the mean of the deviations

Step 6:  Calculate the mean as

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The relation between the Mean of deviations and mean

$d_i=x_i-a$
Summing over all $x_i^{\prime }s,$
$\sum d_i=\sum x_i-\sum a$
Dividing throughout by $\sum f_i=n,$ Where ${}^{}{n}^{\prime }$ is the total number of observations.

For More Information On The Relation Between Mean of deviations and Mean, Watch The Below Video.

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Step-Deviation method of finding mean

Step 1: Classify the data into intervals and find the corresponding frequency of each class.

Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.

Step 3: Take one of the $x_i^{\prime }s$ (usually one in the middle) as assumed mean and denote it by ${}^{}{a}^{\prime }$.

Step 4: Find the deviation of $a$ from each of the $x_i^{\prime }s$
$d_i=x_i-a$

Step 5: Divide all deviations $-d_i$ by the class width (h) to get $u_i^{\prime }s.$

Step 6: Find the mean of $u_i^{\prime }s$

Step 7:  Calculate the mean as

Relation between mean of Step- Deviations (u) and mean

Important relations between methods of finding mean

• All three methods of finding mean yield the same result.
• Step deviation method is easier to apply if all the deviations have a common factor.
• Assumed mean method and step deviation method are simplified versions of the direct method.

Median

Finding the Median of Grouped Data when class Intervals are not given

Step 1: Tabulate the observations and the corresponding frequency in ascending or descending order.

Step 2: Add the cumulative frequency column to the table by finding the cumulative frequency up to each observation.

Step 3: If the number of observations is odd, the median is the observation whose cumulative frequency is just greater than or equal to (n+1)/2

If the number of observations is even, the median is the average of observations whose cumulative frequency is just greater than or equal to n/2 and $(n/2)+1$.

For More Information On Median, Watch The Below Video.

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Cumulative Frequency

Cumulative frequency is obtained by adding all the frequencies up to a certain point.

Finding median for Grouped Data when class Intervals are given

Step 1: find the cumulative frequency for all class intervals.
Step 2: the median class is the class whose cumulative frequency is greater than or nearest to $\frac{n}{2},$ where n is the number of observations.
Step 3: $Media\mathrm{n\; =}l\; +\; [(N/2\; –\; cf)/f]\; \times \; h$

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).

Cumulative Frequency distribution of less than type

Cumulative frequency of the less than type indicates the number of observations which are less than or equal to a particular observation.

Cumulative Frequency distribution of more than type

Cumulative frequency of more than type indicates the number of observations that are greater than or equal to a particular observation.

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Visualising formula for median graphically

Step 1: Identify the median class.

Step 2: Mark cumulative frequencies on the y-axis and observations on the x-axis corresponding to the median class.

Step 3: Draw a straight line graph joining the extremes of class and cumulative frequencies.

Step 4: Identify the point on the graph corresponding to $cf$= n/2

Step 5: Drop a perpendicular from this point onto the x-axis.

Ogive of less than type

The graph of a cumulative frequency distribution of the less than type is called an ‘ogive of the less than type’.

Ogive of more than type

The graph of a cumulative frequency distribution of the more than type is called an ‘ogive of the more than type’.

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Relation between the less than and more than type curves

The point of intersection of the ogives of more than and less than types gives the median of the grouped frequency distribution.

Mode

Finding mode for Grouped Data when class intervals are not given

In grouped data without class intervals, the observation having the largest frequency is the mode.

Finding mode for Ungrouped Data

For ungrouped data, the mode can be found out by counting the observations and using tally marks to construct a frequency table.
The observation having the largest frequency is the mode.

Finding mode for Grouped Data when class intervals are given

For, grouped data, the class having the highest frequency is called the modal class. The mode can be calculated using the following formula. The formula is valid for equal class intervals and when the modal class is unique.
$Mode=\; l\; +\; [(f1\; –\; f0)/(2f1\; –\; f0\; –\; f2)]\; \times \; h$

Where,
$l=$ lower limit of modal class
$h=$ class width
$f_1=$ frequency of the modal class
$f_0=$ frequency of the class preceding the modal class
$f_2=$ frequency of the class succeeding the modal class.

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Visualising formula for mode graphically

Step 1: Express the class intervals and frequencies as a histogram.

Step 2: Join the top corners of the modal class to the diagonally opposite corners of the adjacent classes

Step 3: Drop a perpendicular from the point of intersection of the above on the horizontal x-axis.

Measures of Central Tendency for Grouped Data

i) Mean is the average of a set of observations.
ii) Median is the middle value of a set of observations.
iii) A mode is the most common observation.

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The best-suited measure of central tendency in different cases and the Empirical relationship between them

i) The mean takes into account all the observations and lies between the extremes. It enables us to compare distributions.

ii) In problems where individual observations are not important, and we wish to find out a ‘typical’  observation where half the observations are below and half the observations are above, the median is more appropriate. Median disregards extreme values.

iii) In situations that require establishing the most frequent value or most popular item, the mode is the best choice.

Mean, mode and median are connected by the empirical relationship

3 Median = Mode + 2 Mean

Solved Example

Question:

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Solution:

Let us find the mean of the given data.

Mean = a + (∑f_id_i/∑f_i)

= 135 + (140/68)

= 135 + 2.05

= 137.05

Now, we need to find the cumulative frequency for the given data.

N = 68

N/2 = 68/2 = 34

Cumulative frequency greater than and nearer to 34 is 42 which lies in the interval 125 – 145.

Median class: 125 – 145

Lower limit of the median class = l = 125

Frequency of the median class = f = 20

Cumulative frequency of the class preceding the median class = cf = 22

Class height = h = 20

Median = l + [(N/2 – cf)/f] × h

= 125 + [(34 – 22)/20] × 20

= 125 + 12

= 137

Let us find the mode of the given data.

Highest frequency = 20

Thus, modal class: 125 – 145

Lower limit of the modal class = l = 125

Frequency of modal class = f₁ = 20

Frequency of the class preceding the modal class = f₀ = 13

Frequency of the class succeeding the modal class = f₂ = 14

Class height = h = 20

Mode = l + [(f₁ – f₀)/(2f₁ – f₀ – f₂)] × h

= 125 + [(20 – 13)/ (2 × 20 – 13 – 14)] × 20

= 125 + [(7/(40 – 27)] × 20

= 125 + (140/13)

= 125 + 10.77

= 135.77

Therefore,

Mean = 137.05

Median = 137

Mode = 135.77

Practice Questions

1. The distribution below gives the weights of 30 students of a class. Find the median weight of the students and mark on Ogive curve.

2. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data.

3. Consider the following distribution of daily wages of 50 workers of a factory.

Find the mean daily wages of the workers of the factory by using an appropriate method.