Class 11 Statistics Notes Chapter 5: Measures of Central Tendency

Quick Overview

Measures of Central Tendency are statistical tools used to represent a large set of data with a single value. This value shows the central or typical value of the data.

Need for Central Tendency

• Simplifies large data sets.
• Makes comparison easier.
• Helps in decision-making.
• Represents the whole data through one value.

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Types of Averages

1. Arithmetic Mean (A.M.)

Definition

Arithmetic Mean is obtained by dividing the sum of all observations by the total number of observations.

Formula (Individual Series)

$$\bar{X}=\frac{\Sigma X}{N}$$Xˉ=NΣX​

Where:

• ΣX = Sum of observations
• N = Number of observations

Example

Data: 10, 20, 30, 40, 50$$\bar{X}=\frac{10+20+30+40+50}{5}=30$$Xˉ=510+20+30+40+50​=30

Methods of Finding Mean

A. Direct Method

$$\bar{X}=\frac{\Sigma X}{N}$$Xˉ=NΣX​

Used when data is small and calculations are simple.

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B. Assumed Mean Method

Used when observations are large.

Formula:$$\bar{X}=A+\frac{\Sigma d}{N}$$Xˉ=A+NΣd​

Where:

• A = Assumed Mean
• d = X − A

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C. Step Deviation Method

Used when deviations are large.

Formula:$$\bar{X}=A+\left(\frac{\Sigma d’}{N}\right)\times c$$Xˉ=A+(NΣd′​)×c

Where:$$d’=\frac{X-A}{c}$$d′=cX−A​

• c = Common factor

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Mean for Discrete Series

Formula:$$\bar{X}=\frac{\Sigma fX}{\Sigma f}$$Xˉ=ΣfΣfX​

Where:

• f = Frequency
• X = Value

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Mean for Continuous Series

Steps

1. Find class midpoint (m).
2. Multiply frequency by midpoint.
3. Calculate Σfm.
4. Apply:

$$\bar{X}=\frac{\Sigma fm}{\Sigma f}$$Xˉ=ΣfΣfm​

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Weighted Arithmetic Mean

Used when different items have different importance.

Formula

$$\bar{X}_w=\frac{\Sigma WX}{\Sigma W}$$Xˉw​=ΣWΣWX​

Where:

• W = Weight
• X = Value

Uses

• Price Index Numbers
• Examination Marks
• Economic Analysis

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Properties of Arithmetic Mean

Advantages

✔ Easy to calculate

✔ Based on all observations

✔ Useful for comparison

Limitations

✘ Affected by extreme values

✘ Not suitable for highly skewed data

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Important Property

$$\Sigma(X-\bar{X})=0$$Σ(X−Xˉ)=0

The sum of deviations from the mean is always zero.

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2. Median

Definition

Median is the middle value of an arranged data set.

It divides data into two equal parts.

• 50% observations lie below it.
• 50% observations lie above it.

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Finding Median (Individual Series)

Step 1

Arrange data in ascending order.

Step 2

Find position:$$\frac{N+1}{2}$$2N+1​

Example

Data:
1, 3, 5, 7, 9

Median = 5

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Median for Even Number of Observations

Take average of two middle values.

Example

2, 4, 6, 8

Median:$$\frac{4+6}{2}=5$$24+6​=5

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Median for Discrete Series

Formula

Position of Median:$$\frac{N+1}{2}$$2N+1​

Steps

1. Calculate cumulative frequency.
2. Locate median position.
3. Corresponding value is the median.

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Median for Continuous Series

Formula

$$Median=L+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$$Median=L+(f2N​−cf​)×h

Where:

• L = Lower limit of median class
• cf = Cumulative frequency before median class
• f = Frequency of median class
• h = Class interval

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Advantages of Median

✔ Not affected by extreme values

✔ Useful for income and wealth data

✔ Suitable for open-ended classes

Limitations

✘ Does not use all observations

✘ Difficult in algebraic calculations

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Quartiles

Quartiles divide data into four equal parts.

Types

First Quartile (Q₁)

25% observations lie below it.

Second Quartile (Q₂)

Median.

Third Quartile (Q₃)

75% observations lie below it.

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Formula

Q₁

$$Q_1=\frac{N+1}{4}$$Q1​=4N+1​

Q₃

$$Q_3=\frac{3(N+1)}{4}$$Q3​=43(N+1)​

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Percentiles

Percentiles divide data into 100 equal parts.

Examples:

• P25 = 25th percentile
• P50 = Median
• P90 = 90th percentile

Uses

• Competitive Exams
• Ranking Systems
• Performance Analysis

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3. Mode

Definition

Mode is the value that occurs most frequently in a data set.

Symbol: Mo

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Example

Data:
2, 3, 4, 4, 4, 5, 6

Mode = 4

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Types of Mode

Unimodal

One mode.

Example:
1, 2, 2, 3

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Bimodal

Two modes.

Example:
1, 2, 2, 3, 3

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Multimodal

More than two modes.

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Mode in Discrete Series

The value corresponding to the highest frequency is the mode.

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Mode in Continuous Series

Formula

$$Mode=L+\left(\frac{D_1}{D_1+D_2}\right)\times h$$Mode=L+(D1​+D2​D1​​)×h

Where:

• L = Lower limit of modal class
• D₁ = Difference between modal class frequency and preceding frequency
• D₂ = Difference between modal class frequency and succeeding frequency
• h = Class interval

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Advantages of Mode

✔ Easy to understand

✔ Useful for qualitative data

✔ Represents most common value

Limitations

✘ May not exist

✘ May have more than one value

✘ Not based on all observations

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Comparison of Mean, Median and Mode

Basis	Mean	Median	Mode
Definition	Arithmetic Average	Middle Value	Most Frequent Value
Uses All Observations	Yes	No	No
Affected by Extreme Values	Yes	No	No
Easy to Calculate	Yes	Yes	Yes
Suitable for Qualitative Data	No	Limited	Yes

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Relationship among Mean, Median and Mode

For a moderately skewed distribution:$$Mode = 3Median – 2Mean$$Mode=3Median−2Mean

Generally:$$Mean > Median > Mode$$Mean>Median>Mode

or$$Mean < Median < Mode$$Mean<Median<Mode

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Key Exam Points

Arithmetic Mean

• Most commonly used average.
• Based on all observations.
• Affected by extreme values.

Median

• Positional average.
• Best for skewed distributions.
• Not affected by extreme values.

Mode

• Most frequently occurring value.
• Suitable for qualitative data.
• Used in market research and fashion studies.

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One-Page Revision

Mean

$$\bar{X}=\frac{\Sigma X}{N}$$Xˉ=NΣX​ $$\bar{X}=\frac{\Sigma fX}{\Sigma f}$$Xˉ=ΣfΣfX​

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Median

Individual Series:$$\frac{N+1}{2}$$2N+1​

Continuous Series:$$Median=L+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$$Median=L+(f2N​−cf​)×h

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Quartiles

$$Q_1=\frac{N+1}{4}$$Q1​=4N+1​ $$Q_3=\frac{3(N+1)}{4}$$Q3​=43(N+1)​

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Mode

$$Mode=L+\left(\frac{D_1}{D_1+D_2}\right)\times h$$Mode=L+(D1​+D2​D1​​)×h

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Important Property

$$\Sigma(X-\bar{X})=0$$

Relationship

$$Mode = 3Median – 2Mean$$

Important MCQs, Fill in the Blanks, True/False, Assertion-Reason, Very Short & Short Answer Questions

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A. Multiple Choice Questions (MCQs)

1. Measures of central tendency are used to:

a) Classify data
b) Summarise data
c) Collect data
d) Tabulate data

Ans: b) Summarise data

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2. Which is the most commonly used measure of central tendency?

a) Median
b) Mode
c) Arithmetic Mean
d) Quartile

Ans: c) Arithmetic Mean

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3. Arithmetic Mean is:

a) Positional average
b) Frequency average
c) Simple average
d) Geometric average

Ans: c) Simple average

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4. Arithmetic Mean is calculated by:

a) ΣX × N
b) ΣX ÷ N
c) N ÷ ΣX
d) ΣX − N

Ans: b) ΣX ÷ N

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5. Which average is affected most by extreme values?

a) Median
b) Mode
c) Arithmetic Mean
d) Quartile

Ans: c) Arithmetic Mean

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6. The sum of deviations from Arithmetic Mean is:

a) 1
b) 0
c) N
d) Infinity

Ans: b) 0

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7. Median is:

a) Most frequent value
b) Middle value
c) Largest value
d) Smallest value

Ans: b) Middle value

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8. Median divides a distribution into:

a) 3 equal parts
b) 4 equal parts
c) 2 equal parts
d) 5 equal parts

Ans: c) 2 equal parts

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9. Mode is:

a) Middle value
b) Average value
c) Most frequently occurring value
d) Smallest value

Ans: c) Most frequently occurring value

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10. Which average is best for qualitative data?

a) Mean
b) Median
c) Mode
d) Quartile

Ans: c) Mode

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11. Quartiles divide the data into:

a) 2 parts
b) 3 parts
c) 4 parts
d) 5 parts

Ans: c) 4 parts

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12. Percentiles divide the data into:

a) 10 parts
b) 50 parts
c) 100 parts
d) 25 parts

Ans: c) 100 parts

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13. The second quartile is:

a) Q1
b) Q2
c) Q3
d) P50

Ans: b) Q2

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14. Q2 is equal to:

a) Mean
b) Median
c) Mode
d) Percentile

Ans: b) Median

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15. P50 is:

a) Mean
b) Median
c) Mode
d) Quartile

Ans: b) Median

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16. Which average is least affected by extreme observations?

a) Mean
b) Median
c) Weighted Mean
d) Arithmetic Mean

Ans: b) Median

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17. Weighted Mean is used when:

a) Values are equal
b) Importance differs
c) Data is qualitative
d) No frequency exists

Ans: b) Importance differs

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18. A distribution having one mode is called:

a) Bimodal
b) Multimodal
c) Unimodal
d) Symmetrical

Ans: c) Unimodal

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19. A distribution having two modes is:

a) Unimodal
b) Bimodal
c) Multimodal
d) Skewed

Ans: b) Bimodal

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20. Median is a:

a) Positional Average
b) Mathematical Average
c) Geometric Average
d) Harmonic Average

Ans: a) Positional Average

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B. Fill in the Blanks

1. Arithmetic Mean is represented by ______.
   Ans: X̄
2. Median is the ______ value of an ordered series.
   Ans: middle
3. Mode is the ______ occurring value.
   Ans: most frequently
4. Quartiles divide data into ______ equal parts.
   Ans: four
5. Percentiles divide data into ______ equal parts.
   Ans: hundred
6. The sum of deviations from Arithmetic Mean is always ______.
   Ans: zero
7. Q2 is also known as ______.
   Ans: Median
8. P50 is equal to ______.
   Ans: Median
9. Arithmetic Mean is affected by ______ values.
   Ans: extreme
10. Weighted Mean uses ______ according to importance.
    Ans: weights
11. Median is not affected by ______ values.
    Ans: extreme
12. Mode is suitable for ______ data.
    Ans: qualitative
13. The first quartile is represented by ______.
    Ans: Q1
14. The third quartile is represented by ______.
    Ans: Q3
15. Arithmetic Mean uses ______ observations.
    Ans: all

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C. True or False

1. Arithmetic Mean is affected by extreme values.
   Ans: True
2. Median is affected by extreme values.
   Ans: False
3. Mode is the most frequent observation.
   Ans: True
4. Median divides data into two equal parts.
   Ans: True
5. Quartiles divide data into four equal parts.
   Ans: True
6. P50 equals Median.
   Ans: True
7. Mean is a positional average.
   Ans: False
8. Median is a positional average.
   Ans: True
9. Mode always exists.
   Ans: False
10. Weighted Mean assigns importance to observations.
    Ans: True
11. Q2 and Median are the same.
    Ans: True
12. Mean is based on all observations.
    Ans: True
13. Median is based on all observations.
    Ans: False
14. Mode is useful for fashion studies.
    Ans: True
15. Percentiles divide data into 100 parts.
    Ans: True

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D. Match the Following

Column A	Column B
Arithmetic Mean	Sum of observations ÷ Number of observations
Median	Middle value
Mode	Most frequent value
Q1	First Quartile
Q3	Third Quartile
P50	Median
Weighted Mean	Uses weights
Quartiles	Four equal parts
Percentiles	Hundred equal parts
Unimodal	One mode

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E. Assertion and Reason Questions

1.

Assertion (A): Arithmetic Mean is affected by extreme values.

Reason (R): Arithmetic Mean uses all observations.

Ans: Both A and R are true and R is the correct explanation.

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2.

Assertion (A): Median is suitable for income data.

Reason (R): Median is not affected by extreme values.

Ans: Both A and R are true and R is the correct explanation.

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3.

Assertion (A): Mode is useful for qualitative data.

Reason (R): Mode identifies the most frequent category.

Ans: Both A and R are true and R is the correct explanation.

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4.

Assertion (A): Quartiles divide data into four equal parts.

Reason (R): Q2 is Median.

Ans: Both A and R are true but R is not the correct explanation.

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F. One Word Answer Questions

1. Most commonly used average?
   Ans: Arithmetic Mean
2. Middle value of ordered data?
   Ans: Median
3. Most frequent value?
   Ans: Mode
4. Second Quartile?
   Ans: Median
5. P50?
   Ans: Median
6. Average using weights?
   Ans: Weighted Mean
7. Quartile below which 25% values lie?
   Ans: Q1
8. Quartile below which 75% values lie?
   Ans: Q3
9. Distribution with one mode?
   Ans: Unimodal
10. Distribution with two modes?
    Ans: Bimodal

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G. Very Short Answer Questions (1 Mark)

1. Define Arithmetic Mean.
2. Define Median.
3. Define Mode.
4. What is Weighted Mean?
5. What is Quartile?
6. What is Percentile?
7. State one property of Arithmetic Mean.
8. Why is Median called a positional average?
9. Which average is suitable for qualitative data?
10. What is P50?

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H. Short Answer Questions (3 Marks)

1. Explain Arithmetic Mean.
2. State any three advantages of Arithmetic Mean.
3. Explain Median with an example.
4. State advantages of Median.
5. Explain Mode and its uses.
6. Differentiate between Mean and Median.
7. Explain Quartiles.
8. Explain Percentiles.
9. What is Weighted Mean? Give one example.
10. Distinguish between Q1, Q2 and Q3.

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I. Long Answer Questions (5 Marks)

1. Explain Measures of Central Tendency.
2. Discuss Arithmetic Mean and its methods of calculation.
3. Explain Median and its computation.
4. Explain Mode and its significance.
5. Compare Mean, Median and Mode.
6. Explain Quartiles and Percentiles.
7. Discuss merits and limitations of Arithmetic Mean.
8. Why is Median preferred in income distribution?
9. Explain Weighted Arithmetic Mean with examples.
10. Describe the relationship among Mean, Median and Mode.

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Important Formula-Based Questions

Arithmetic Mean

$$\bar{X}=\frac{\Sigma X}{N}$$Xˉ=NΣX​

Discrete Series Mean

$$\bar{X}=\frac{\Sigma fX}{\Sigma f}$$Xˉ=ΣfΣfX​

Median Position

$$\frac{N+1}{2}$$2N+1​

First Quartile

$$Q_1=\frac{N+1}{4}$$Q1​=4N+1​

Third Quartile

$$Q_3=\frac{3(N+1)}{4}$$Q3​=43(N+1)​

Continuous Series Median

$$Median=L+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$$Median=L+(f2N​−cf​)×h

Mode

$$Mode=L+\left(\frac{D_1}{D_1+D_2}\right)\times h$$Mode=L+(D1​+D2​D1​​)×h

Relationship

$$Mode = 3Median – 2Mean$$Mode=3Median−2Mean