What are the different measures of central tendency?

The main measures of central tendency are: Mean (Average): The sum of all data points divided by the number of points. Median: The middle value when data points are arranged in ascending or descending order. Mode: The most frequently occurring value in a dataset.

When should I use the mean?

You should use the mean (average) when you want to find the overall average of a dataset and there are no extreme values (outliers) that could skew the result. The mean helps get a general idea of your data's "central" value when all values are similar. For example, if you want to know the average test score of a class where most students scored similarly, the mean is a good measure to use.

When should I use the median?

The median is used when you need the middle value of a dataset, especially if there are outliers. It’s ideal for skewed data, like income, to avoid distortion by extreme values.

When should I use the mode?

The mode is ideal for categorical data or when identifying the most common value in a dataset. It is beneficial when data is not numerical.

How is the weighted mean different from the mean?

The weighted mean assigns different weights to data points based on their importance, while the standard mean treats all data points equally. The weighted mean is calculated by multiplying each data point by its corresponding weight, adding these products together, and then dividing by the total sum of the weights.

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Central Tendency

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Statistics: An Overview

Statistics is a mathematical branch that deals with collecting, analyzing, interpreting, presenting, and organizing data.

Probability Formulas and Bayes Theorem

Probability, the science of possibility, resides within the realm of mathematics, navigating the unpredictable waters of random events.

Geometric Mean

In mathematics and statistics, summarizing a data set can be effectively achieved using measures of central tendency.

Sets

In Mathematics, a set is a well-defined collection of different objects, considered as an object in its own right.

Binomial Theorem

The Binomial Theorem is an essential tool for simplifying long expressions that follow the pattern

Continuity and Discontinuity

Continuity in mathematics means a function has no interruptions at a point, meeting three conditions: defined at the point, limit exists

Central Tendency

Q: What is the central tendency in statistics?

Central tendency is a fundamental concept in statistics that describes the center point or typical value of a dataset. It helps in summarizing data sets with a single value that represents the entire data. The three main measures of central tendency include mean, median, and mode.

Central Tendency is a fundamental concept in statistics that describes the center point or typical value of a dataset. It helps in summarizing data sets with a single value that represents the entire data. The three main measures of central tendency are:

Mean: The average of all data points, calculated by adding them up and dividing them by the number of points.
Median: The middle value in an ordered dataset. It is a measure of central tendency that is not affected by outliers.
Mode: The most frequently occurring value in a dataset, highlighting the most common data point.

1.0Central Tendency Definition

Central Tendency in Statistics is a concept used to identify the central or average value within a data set. It provides a single value that represents the typical or central value of the data distribution. The main three measures of central tendency are the mean, median, and mode.

2.0Types of Central Tendency

The three main types of Central Tendency are:

1. Mean: Also known as the average, the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

2. Median: The median is determined as the central value within a dataset once the values have been arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. It is less affected by extreme values and is useful for skewed distributions.

3. Mode: The mode is the value that appears most often in the data set.It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

These measures offer important insights into the central tendencies of a dataset, helping researchers and analysts understand its characteristics and make informed decisions.

Arithmetic Mean

For Ungrouped Distribution

The arithmetic mean, often simply called the mean, is a measure of central tendency of statistics that represents the average of a set of numerical values. It is calculated by adding up all the values in the data set and then dividing the sum by the total number of values.

The formula for calculating the arithmetic $m e an (\overline{x})$ of a data set x₁, x₂, ..., x_n with n observations is:

$\overline{x} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$

In simpler terms, you add up all the values and then divide by how many values there are. The mean provides a single value that summarizes the central tendency of the data set. It is widely used in various fields for summarizing and analyzing data. However, it can be influenced by extreme values (outliers) in the data set, so it's important to consider the context of the data when interpreting the mean.

For Ungrouped Frequency Distribution

For ungrouped frequency distributions, where each data point has a corresponding frequency (how often it occurs), the arithmetic mean is calculated slightly differently.

Let's say we have a set of n data points x₁, x₂, ..., x_n with corresponding frequencies f₁, f₂, ..., f_n. To find the $m e an (\overline{x})$ , you follow these steps:

1. Multiply each data point by its respective frequency.

2. Sum up all these products.

3. Divide the sum by the total frequency (sum of all frequencies).

The formula for calculating the mean for ungrouped frequency distributions is:

$\overline{x} = \frac{f _{1} x _{1} + f _{2} x _{2} + \dots . + f _{n} x _{n}}{( f _{1} + f _{2} + \dots .. + f _{n} )}$

$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}} or \overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{N} Where N = \sum_{i = 1}^{n} f_{i}$

In other words, you are finding the weighted average of the data points, where each data point is weighted by its frequency. This accounts for the fact that some values may occur more frequently than others in the data set.

Two Interesting Properties of A.M.

The sum of deviations of data points from the arithmetic mean is always zero. Mathematically, this is represented as $\sum (x - \overline{x}) = 0$ .
The arithmetic mean is sensitive to extreme values. Any significantly large or small value can skew the mean upward or downward.

Weighted Mean (W.M.)

If $x_{1}, x_{2}, \dots\dots, x_{n}$ are n values of variate $\overline{X}$ and their weightages are $w_{1}, w_{2}, \dots\dots, w_{n}$ respectively, then their weighted mean is -

$W.M. = \frac{w _{1} x _{1} + w _{2} x _{2} + \dots\dots . + w _{n} x _{n}}{w _{1} + w _{2} + \dots\dots . + w _{n}} = \frac{\sum _{i = 1}^{n} w _{i} x _{i}}{\sum _{i = 1}^{n} w _{i}}$

Median

The median serves as an additional measure of central tendency, similar to the mean, but it represents the middle value of a data set when arranged in ascending or descending order. To find the median:

Arrange the data set in either ascending or descending order.
When the number of observations (n) is odd, the median corresponds to the value positioned at the center of the ordered list.
If the total number of observations (n) is even, the median is calculated as the arithmetic mean of the two middle values.
If n is the number of observations:
For Ungrouped Distribution and Ungrouped Frequency Distribution

For odd $n : Median = Value of (\frac{n + 1}{2})^{t h} observation.$
For even $n : Median = \frac{Value of ( \frac{n}{2} ) ^{t h} observation + Value of ( \frac{n}{2} + 1 ) ^{t h} observation}{2}$ .

The median is often preferred over the mean when the data set contains outliers or is skewed, as it is less affected by extreme values. It provides a better representation of the typical value in such cases.

For Grouped Frequency Distribution

When dealing with a grouped frequency distribution, where data is organized into intervals or classes along with their corresponding frequencies, finding the median involves a slightly different approach.

Here's how to find the median for a grouped frequency distribution:

Find the Median Class: Determine the class interval containing the median. This is usually the class where the cumulative frequency exceeds half of the total frequency.
Calculate Median: Use the formula for finding the median within a class interval:
$Median = l + \frac{( \frac{N}{2} - F )}{f} \times h$

where:

l = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency (c. f) of the class before the median class

f = Frequency of the class interval that contains the median.

h = Width of the class interval

This formula calculates the median by finding a value within the median class, considering the width of the class interval and the cumulative frequencies.

If the median class is not a continuous interval (e.g., if it's a single value), you can still use finding a value within that interval using similar principles.

Once you find the median, it represents the middle value of the data set. It's a useful measure of central tendency, especially for data sets with continuous values.

Mode

Mode is the value of the variate, which has maximum Frequency.

(i) for ungrouped distribution

The value of the variate which repeated the maximum number of times.

(ii) for ungrouped frequency distribution.

The value of the variate which has the maximum frequency.

(iii) for grouped frequency distribution.

First, we find the class with the maximum frequency. This is a modal class, then

$Mode = I + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$

Where l = lower limit of modal class.

f₀ = frequency of modal class.

f₁ = frequency of the class preceding modal class.

f₂ = frequency of the class succeeding modal class.

h = class interval of modal class.

3.0Relation Between Central Tendency

In a moderately asymmetrical distribution following relation between mean, median and mode of a distribution.

It is known as the Empirical Formula.

Mode = 3 Median – 2 Mean

4.0Relative Position of Mean, Median, Mode

Relative Position of Arithmetic Mean(Mean), Median and Mode:-

If we denote:

Arithmetic Mean = M_e

Median = M_i

Mode = M_o

Then, their relative magnitudes are such that either M_e > M_i > M_o or M_e < M_i < M_o (with suffixes occurring in alphabetical order). It's noteworthy that the median always falls between the arithmetic mean and the mode.

5.0Solved Examples

Example 1: Mean of n observation $1^{2}, 2^{2}, 3^{2}, \dots, n^{2} is \frac{46 n}{11}$ . Then value of n is:

(A) 11 (B) 12 (C) 23 (D) 22

Ans. (A)

Solution:

$\overset{x}{ˉ} = \frac{\sum x _{i}}{n} \Rightarrow \frac{46 n}{11} = \frac{1 ^{2} + 2 ^{2} + \dots\dots n ^{2}}{n}$

$\Rightarrow \frac{46 n}{11} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6 n}$

Solving

n = 11

Example 2: Find the A.M of the following frequency distribution

_{Wage(in Rs.)}	₈₀₀	₈₂₀	₈₆₀	₉₀₀	₉₂₀	₉₈₀	₁₀₀₀
_{No. of Worker}	₇	₁₄	₁₉	₂₄	₂₀	₁₁	₅

Solution:

Let a = 900, h = 20

$x_{i}$	$f_{i}$	$d_{i} = x_{i}$	$u_{i} = \frac{d _{i}}{h}$	$f_{i} u_{i}$
800	7	-100	-5	-35
820	14	-80	-4	-56
860	19	-40	-2	-38
900	24	0	0	0
920	20	20	1	20
980	11	80	4	44
1000	5	100	5	25
	N = 100			$\sum f_{i} u_{i} = - 40$

∵ $\overset{x}{ˉ} = a + \frac{\sum f _{i} u _{i}}{N} \times h$

₌ $900 + \frac{( - 40 )}{100} \times 20$

= 900 – 8

= 892

Example 3: Find the weighted mean of first n natural numbers when their weights are equal to their squares respectively

Solution:

$W e i g h t e d M e an = \frac{1. 1 ^{2} + 2. 2 ^{2} + \dots . + n \cdot n ^{2}}{1 ^{2} + 2 ^{2} + \dots . + n ^{2}} = \frac{1 ^{3} + 2 ^{3} + \dots . + n ^{3}}{1 ^{2} + 2 ^{2} + \dots . + n ^{2}} = \frac{[ n ( n + 1 ) /2 ] ^{2}}{[ n ( n + 1 ) ( 2 n + 1 ) /6 ]} = \frac{3 n ( n + 1 )}{2 ( 2 n + 1 )}$

Example 4: Find the median of the following frequency distribution.

_Class

_1-10

_11-20

_21-30

_31-40

_41-50

$f_{i}$

₅

₆

₈

₇

₄

Solution:

Class	$f_{i}$	C.f.
0.5-10.5	5	5
10.5-20.5	6	11
20.5-30.5	8	19
30.5-40.5	7	26
40.5-50.5	4	30

Here $\frac{N}{2} = \frac{30}{2} = 15$

Median class is 20.5 – 30.5

∴ Median = $20.5 + \frac{( 15 - 11 )}{8} \times 10$

= 20.5 + 5 = 25.5

Example 5: Find the mode of the following frequency distribution.

$x_{i}$

₄

₅

₆

₇

₈

₉

$f_{i}$

₃

₄

₅

₈

₇

₆

Solution:

Here the variate 7 has maximum frequency.

∴ Mode = 7

Example 6: Find the mode of following frequency distribution.

_Class

_0-10

_10-20

_20-30

_30-40

_40-50

$f_{i}$

₈

₃₀

₄₀

₁₀

₁₂

Solution:

Here model class is 20 – 30 (which have maximum frequency)

$Mode = l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

= $Mode = l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

= 20 + 2.5

= 22.5

6.0Practice Questions

1. If the mean of the numbers 27 + x, 31 + x, 89 + x, 107 + x,156 + x is 82, then the mean of

130 + x, 126 + x, 68 + x, 50 + x, 1 + x is

(A) 75 (B) 157 (C) 82 (D) 80

2. Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is

(A) 48 (B) 82 $\frac{1}{2}$ (C) 50 (D) 80

3. The mean weight per student in a group of seven students is 55 kg. If the individual weights of 6 students are 52, 58, 55, 53, 56 and 54; then weights of the seventh student is

(A) 55 kg (B) 60 kg (C) 57 kg (D) 50 kg

4. The mode of the distribution

Marks	4	5	6	7	8
No. of students	6	7	10	8	3

(A) 5 (B) 6 (C) 8 (D) 10

5. If the difference between mean and mode is 63, the difference between mean and median is

(A) 189 (B) 21 (C) 31.5 (D) 48.5

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Statistics: An Overview

Statistics is a mathematical branch that deals with collecting, analyzing, interpreting, presenting, and organizing data.

Probability Formulas and Bayes Theorem

Probability, the science of possibility, resides within the realm of mathematics, navigating the unpredictable waters of random events.

Geometric Mean

In mathematics and statistics, summarizing a data set can be effectively achieved using measures of central tendency.

Sets

In Mathematics, a set is a well-defined collection of different objects, considered as an object in its own right.

Binomial Theorem

The Binomial Theorem is an essential tool for simplifying long expressions that follow the pattern

Continuity and Discontinuity

Continuity in mathematics means a function has no interruptions at a point, meeting three conditions: defined at the point, limit exists

Central Tendency

Central Tendency is a fundamental concept in statistics that describes the center point or typical value of a dataset. It helps in summarizing data sets with a single value that represents the entire data. The three main measures of central tendency are:

Mean: The average of all data points, calculated by adding them up and dividing them by the number of points.
Median: The middle value in an ordered dataset. It is a measure of central tendency that is not affected by outliers.
Mode: The most frequently occurring value in a dataset, highlighting the most common data point.

1.0Central Tendency Definition

Central Tendency in Statistics is a concept used to identify the central or average value within a data set. It provides a single value that represents the typical or central value of the data distribution. The main three measures of central tendency are the mean, median, and mode.

2.0Types of Central Tendency

The three main types of Central Tendency are:

1. Mean: Also known as the average, the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

2. Median: The median is determined as the central value within a dataset once the values have been arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. It is less affected by extreme values and is useful for skewed distributions.

3. Mode: The mode is the value that appears most often in the data set.It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

These measures offer important insights into the central tendencies of a dataset, helping researchers and analysts understand its characteristics and make informed decisions.

Arithmetic Mean

For Ungrouped Distribution

The arithmetic mean, often simply called the mean, is a measure of central tendency of statistics that represents the average of a set of numerical values. It is calculated by adding up all the values in the data set and then dividing the sum by the total number of values.

The formula for calculating the arithmetic $m e an (\overline{x})$ of a data set x₁, x₂, ..., x_n with n observations is:

$\overline{x} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$

In simpler terms, you add up all the values and then divide by how many values there are. The mean provides a single value that summarizes the central tendency of the data set. It is widely used in various fields for summarizing and analyzing data. However, it can be influenced by extreme values (outliers) in the data set, so it's important to consider the context of the data when interpreting the mean.

For Ungrouped Frequency Distribution

For ungrouped frequency distributions, where each data point has a corresponding frequency (how often it occurs), the arithmetic mean is calculated slightly differently.

Let's say we have a set of n data points x₁, x₂, ..., x_n with corresponding frequencies f₁, f₂, ..., f_n. To find the $m e an (\overline{x})$ , you follow these steps:

1. Multiply each data point by its respective frequency.

2. Sum up all these products.

3. Divide the sum by the total frequency (sum of all frequencies).

The formula for calculating the mean for ungrouped frequency distributions is:

$\overline{x} = \frac{f _{1} x _{1} + f _{2} x _{2} + \dots . + f _{n} x _{n}}{( f _{1} + f _{2} + \dots .. + f _{n} )}$

$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}} or \overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{N} Where N = \sum_{i = 1}^{n} f_{i}$

In other words, you are finding the weighted average of the data points, where each data point is weighted by its frequency. This accounts for the fact that some values may occur more frequently than others in the data set.

Two Interesting Properties of A.M.

The sum of deviations of data points from the arithmetic mean is always zero. Mathematically, this is represented as $\sum (x - \overline{x}) = 0$ .
The arithmetic mean is sensitive to extreme values. Any significantly large or small value can skew the mean upward or downward.

Weighted Mean (W.M.)

If $x_{1}, x_{2}, \dots\dots, x_{n}$ are n values of variate $\overline{X}$ and their weightages are $w_{1}, w_{2}, \dots\dots, w_{n}$ respectively, then their weighted mean is -

$W.M. = \frac{w _{1} x _{1} + w _{2} x _{2} + \dots\dots . + w _{n} x _{n}}{w _{1} + w _{2} + \dots\dots . + w _{n}} = \frac{\sum _{i = 1}^{n} w _{i} x _{i}}{\sum _{i = 1}^{n} w _{i}}$

Median

The median serves as an additional measure of central tendency, similar to the mean, but it represents the middle value of a data set when arranged in ascending or descending order. To find the median:

Arrange the data set in either ascending or descending order.
When the number of observations (n) is odd, the median corresponds to the value positioned at the center of the ordered list.
If the total number of observations (n) is even, the median is calculated as the arithmetic mean of the two middle values.
If n is the number of observations:
For Ungrouped Distribution and Ungrouped Frequency Distribution

For odd $n : Median = Value of (\frac{n + 1}{2})^{t h} observation.$
For even $n : Median = \frac{Value of ( \frac{n}{2} ) ^{t h} observation + Value of ( \frac{n}{2} + 1 ) ^{t h} observation}{2}$ .

The median is often preferred over the mean when the data set contains outliers or is skewed, as it is less affected by extreme values. It provides a better representation of the typical value in such cases.

For Grouped Frequency Distribution

When dealing with a grouped frequency distribution, where data is organized into intervals or classes along with their corresponding frequencies, finding the median involves a slightly different approach.

Here's how to find the median for a grouped frequency distribution:

Find the Median Class: Determine the class interval containing the median. This is usually the class where the cumulative frequency exceeds half of the total frequency.
Calculate Median: Use the formula for finding the median within a class interval:
$Median = l + \frac{( \frac{N}{2} - F )}{f} \times h$

where:

l = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency (c. f) of the class before the median class

f = Frequency of the class interval that contains the median.

h = Width of the class interval

This formula calculates the median by finding a value within the median class, considering the width of the class interval and the cumulative frequencies.

If the median class is not a continuous interval (e.g., if it's a single value), you can still use finding a value within that interval using similar principles.

Once you find the median, it represents the middle value of the data set. It's a useful measure of central tendency, especially for data sets with continuous values.

Mode

Mode is the value of the variate, which has maximum Frequency.

(i) for ungrouped distribution

The value of the variate which repeated the maximum number of times.

(ii) for ungrouped frequency distribution.

The value of the variate which has the maximum frequency.

(iii) for grouped frequency distribution.

First, we find the class with the maximum frequency. This is a modal class, then

$Mode = I + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$

Where l = lower limit of modal class.

f₀ = frequency of modal class.

f₁ = frequency of the class preceding modal class.

f₂ = frequency of the class succeeding modal class.

h = class interval of modal class.

3.0Relation Between Central Tendency

In a moderately asymmetrical distribution following relation between mean, median and mode of a distribution.

It is known as the Empirical Formula.

Mode = 3 Median – 2 Mean

4.0Relative Position of Mean, Median, Mode

Relative Position of Arithmetic Mean(Mean), Median and Mode:-

If we denote:

Arithmetic Mean = M_e

Median = M_i

Mode = M_o

Then, their relative magnitudes are such that either M_e > M_i > M_o or M_e < M_i < M_o (with suffixes occurring in alphabetical order). It's noteworthy that the median always falls between the arithmetic mean and the mode.

5.0Solved Examples

Example 1: Mean of n observation $1^{2}, 2^{2}, 3^{2}, \dots, n^{2} is \frac{46 n}{11}$ . Then value of n is:

(A) 11 (B) 12 (C) 23 (D) 22

Ans. (A)

Solution:

$\overset{x}{ˉ} = \frac{\sum x _{i}}{n} \Rightarrow \frac{46 n}{11} = \frac{1 ^{2} + 2 ^{2} + \dots\dots n ^{2}}{n}$

$\Rightarrow \frac{46 n}{11} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6 n}$

Solving

n = 11

Example 2: Find the A.M of the following frequency distribution

_{Wage(in Rs.)}	₈₀₀	₈₂₀	₈₆₀	₉₀₀	₉₂₀	₉₈₀	₁₀₀₀
_{No. of Worker}	₇	₁₄	₁₉	₂₄	₂₀	₁₁	₅

Solution:

Let a = 900, h = 20

$x_{i}$	$f_{i}$	$d_{i} = x_{i}$	$u_{i} = \frac{d _{i}}{h}$	$f_{i} u_{i}$
800	7	-100	-5	-35
820	14	-80	-4	-56
860	19	-40	-2	-38
900	24	0	0	0
920	20	20	1	20
980	11	80	4	44
1000	5	100	5	25
	N = 100			$\sum f_{i} u_{i} = - 40$

∵ $\overset{x}{ˉ} = a + \frac{\sum f _{i} u _{i}}{N} \times h$

₌ $900 + \frac{( - 40 )}{100} \times 20$

= 900 – 8

= 892

Example 3: Find the weighted mean of first n natural numbers when their weights are equal to their squares respectively

Solution:

$W e i g h t e d M e an = \frac{1. 1 ^{2} + 2. 2 ^{2} + \dots . + n \cdot n ^{2}}{1 ^{2} + 2 ^{2} + \dots . + n ^{2}} = \frac{1 ^{3} + 2 ^{3} + \dots . + n ^{3}}{1 ^{2} + 2 ^{2} + \dots . + n ^{2}} = \frac{[ n ( n + 1 ) /2 ] ^{2}}{[ n ( n + 1 ) ( 2 n + 1 ) /6 ]} = \frac{3 n ( n + 1 )}{2 ( 2 n + 1 )}$

Example 4: Find the median of the following frequency distribution.

_Class

_1-10

_11-20

_21-30

_31-40

_41-50

$f_{i}$

₅

₆

₈

₇

₄

Solution:

Class	$f_{i}$	C.f.
0.5-10.5	5	5
10.5-20.5	6	11
20.5-30.5	8	19
30.5-40.5	7	26
40.5-50.5	4	30

Here $\frac{N}{2} = \frac{30}{2} = 15$

Median class is 20.5 – 30.5

∴ Median = $20.5 + \frac{( 15 - 11 )}{8} \times 10$

= 20.5 + 5 = 25.5

Example 5: Find the mode of the following frequency distribution.

$x_{i}$

₄

₅

₆

₇

₈

₉

$f_{i}$

₃

₄

₅

₈

₇

₆

Solution:

Here the variate 7 has maximum frequency.

∴ Mode = 7

Example 6: Find the mode of following frequency distribution.

_Class

_0-10

_10-20

_20-30

_30-40

_40-50

$f_{i}$

₈

₃₀

₄₀

₁₀

₁₂

Solution:

Here model class is 20 – 30 (which have maximum frequency)

$Mode = l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

= $Mode = l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

= 20 + 2.5

= 22.5

6.0Practice Questions

1. If the mean of the numbers 27 + x, 31 + x, 89 + x, 107 + x,156 + x is 82, then the mean of

130 + x, 126 + x, 68 + x, 50 + x, 1 + x is

(A) 75 (B) 157 (C) 82 (D) 80

2. Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is

(A) 48 (B) 82 $\frac{1}{2}$ (C) 50 (D) 80

3. The mean weight per student in a group of seven students is 55 kg. If the individual weights of 6 students are 52, 58, 55, 53, 56 and 54; then weights of the seventh student is

(A) 55 kg (B) 60 kg (C) 57 kg (D) 50 kg

4. The mode of the distribution

Marks	4	5	6	7	8
No. of students	6	7	10	8	3

(A) 5 (B) 6 (C) 8 (D) 10

5. If the difference between mean and mode is 63, the difference between mean and median is

(A) 189 (B) 21 (C) 31.5 (D) 48.5

Frequently Asked Questions

What is the central tendency in statistics?

What are the different measures of central tendency?

When should I use the mean?

When should I use the median?

When should I use the mode?

How is the weighted mean different from the mean?

Join ALLEN!

Related Articles:-

Statistics: An Overview

Probability Formulas and Bayes Theorem

Geometric Mean

Sets

Binomial Theorem

Continuity and Discontinuity

Central Tendency

1.0Central Tendency Definition

2.0Types of Central Tendency

Arithmetic Mean

Two Interesting Properties of A.M.

Weighted Mean (W.M.)

Median

Mode

3.0Relation Between Central Tendency

4.0Relative Position of Mean, Median, Mode

5.0Solved Examples

6.0Practice Questions

Table of Contents

Frequently Asked Questions

What is the central tendency in statistics?

What are the different measures of central tendency?

When should I use the mean?

When should I use the median?

When should I use the mode?

How is the weighted mean different from the mean?

Join ALLEN!

Related Articles:-

Statistics: An Overview

Probability Formulas and Bayes Theorem

Geometric Mean

Sets

Binomial Theorem

Continuity and Discontinuity

Central Tendency

1.0Central Tendency Definition

2.0Types of Central Tendency

Arithmetic Mean

Two Interesting Properties of A.M.

Weighted Mean (W.M.)

Median

Mode

3.0Relation Between Central Tendency

4.0Relative Position of Mean, Median, Mode

5.0Solved Examples

6.0Practice Questions

Table of Contents