Mean, Median, Mode, and Range - Complete Guide

Welcome to this comprehensive guide on statistical measures of central tendency and dispersion. In this resource, you'll learn about:

Mean (Arithmetic Average)
Median (Middle Value)
Mode (Most Frequent Value)
Range (Measure of Dispersion)

Each concept includes detailed explanations, formulas, step-by-step examples, and special cases. Test your understanding with the interactive quiz at the end!

Mean (Arithmetic Average)

The mean is the most common measure of central tendency, calculated by adding all values in a dataset and dividing by the number of values.

Mean (μ) = (Sum of all values) ÷ (Number of values)
Mean = (x₁ + x₂ + x₃ + ... + xₙ) ÷ n
Mean = Σx ÷ n

Basic Example

Find the mean of: 5, 8, 12, 15, 20

Step 1: Add all the values.

5 + 8 + 12 + 15 + 20 = 60

Step 2: Count the number of values.

n = 5

Step 3: Divide the sum by the count.

Mean = 60 ÷ 5 = 12

Therefore, the mean is 12.

Types of Mean

The arithmetic mean is the standard mean we typically use, as shown in the formula above.

Find the arithmetic mean of: 4, 7, 9, 12, 18

Sum = 4 + 7 + 9 + 12 + 18 = 50

Count = 5

Mean = 50 ÷ 5 = 10

The weighted mean gives different weights (importance) to different values.

Weighted Mean = (w₁x₁ + w₂x₂ + ... + wₙxₙ) ÷ (w₁ + w₂ + ... + wₙ)

A student has these grades: Test 1: 85 (weight 20%), Test 2: 90 (weight 30%), Final Exam: 78 (weight 50%). Find the weighted mean.

Weighted Mean = (0.2 × 85) + (0.3 × 90) + (0.5 × 78)

= 17 + 27 + 39

= 83

The geometric mean is useful for data that represents rates or percentages and is calculated by taking the nth root of the product of n values.

Geometric Mean = ⁿ√(x₁ × x₂ × ... × xₙ)

Find the geometric mean of: 2, 8, 16

Geometric Mean = ³√(2 × 8 × 16)

= ³√256

≈ 6.35

The harmonic mean is useful for rates and ratios, particularly rates of change.

Harmonic Mean = n ÷ (1/x₁ + 1/x₂ + ... + 1/xₙ)

Find the harmonic mean of: 4, 5, 10

Harmonic Mean = 3 ÷ (1/4 + 1/5 + 1/10)

= 3 ÷ (0.25 + 0.2 + 0.1)

= 3 ÷ 0.55

≈ 5.45

Mean with Grouped Data

When dealing with grouped data (data organized in frequency tables), we use a different approach to find the mean.

Mean for Grouped Data = Σ(f × m) ÷ Σf
Where f = frequency and m = midpoint of each class

Find the mean of this grouped data:

Class Interval	Frequency (f)	Midpoint (m)	f × m
1-5	8	3	24
6-10	12	8	96
11-15	15	13	195
16-20	10	18	180
21-25	5	23	115
Total	50		610

Mean = Σ(f × m) ÷ Σf

Mean = 610 ÷ 50

Mean = 12.2

Important Properties of Mean

The mean is affected by every value in the dataset.
The mean is sensitive to outliers (extremely high or low values).
The sum of deviations from the mean equals zero: Σ(x - μ) = 0
The mean minimizes the sum of squared deviations: Σ(x - μ)² is minimized at x = μ

Mean Calculator

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Median (Middle Value)

The median is the middle value of a dataset when arranged in ascending or descending order. It divides the dataset into two equal halves.

How to Find the Median

For an odd number of values:
Median = Value at position (n+1)/2

For an even number of values:
Median = Average of values at positions n/2 and (n/2)+1

Examples of Finding the Median

Find the median of: 7, 13, 4, 10, 6

Step 1: Arrange the data in ascending order.

4, 6, 7, 10, 13

Step 2: Find the middle position.

n = 5 (odd count)

Middle position = (n+1)/2 = (5+1)/2 = 3

Step 3: Find the value at the middle position.

Value at position 3 = 7

Therefore, the median is 7.

Find the median of: 15, 8, 12, 6, 22, 9

Step 1: Arrange the data in ascending order.

6, 8, 9, 12, 15, 22

Step 2: Find the middle positions.

n = 6 (even count)

Middle positions = n/2 and (n/2)+1 = 3 and 4

Step 3: Find the values at these positions and calculate their average.

Values at positions 3 and 4 = 9 and 12

Median = (9 + 12) ÷ 2 = 10.5

Therefore, the median is 10.5.

For grouped data, we use the following formula to estimate the median:

Median = L + [(n/2 - CF) / f] × w
Where:
L = Lower boundary of the median class
n = Total frequency
CF = Cumulative frequency before the median class
f = Frequency of the median class
w = Width of the median class

Find the median of this grouped data:

Class	Frequency (f)	Cumulative Frequency (CF)
10-19	5	5
20-29	12	17
30-39	20	37
40-49	8	45
50-59	5	50

Step 1: Find n/2

Total frequency n = 50

n/2 = 25

Step 2: Identify the median class (where cumulative frequency first exceeds n/2)

Median class = 30-39 (CF before = 17, CF after = 37)

Step 3: Apply the formula

L = 30

CF = 17

f = 20

w = 10

Median = 30 + [(25 - 17) / 20] × 10

= 30 + (8 / 20) × 10

= 30 + 4

= 34

Therefore, the estimated median is 34.

Important Properties of Median

The median is not affected by extreme values or outliers.
The median may not be an actual value in the dataset (especially with even counts).
The median is position-based, not calculation-based.
For skewed distributions, the median is often a better measure of central tendency than the mean.

Median Calculator

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Mode (Most Frequent Value)

The mode is the value that appears most frequently in a dataset. A dataset can have no mode, one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

How to Find the Mode

1. Count the frequency of each value in the dataset.

2. Identify the value(s) with the highest frequency.

Examples of Finding the Mode

Find the mode of: 5, 2, 7, 2, 9, 8, 2, 3, 7

Step 1: Count the frequency of each value.

2 appears 3 times

3 appears 1 time

5 appears 1 time

7 appears 2 times

8 appears 1 time

9 appears 1 time

Step 2: Identify the value with the highest frequency.

2 appears most frequently (3 times)

Therefore, the mode is 2.

Find the mode of: 4, 7, 2, 7, 9, 4, 3, 8, 4, 7

Step 1: Count the frequency of each value.

2 appears 1 time

3 appears 1 time

4 appears 3 times

7 appears 3 times

8 appears 1 time

9 appears 1 time

Step 2: Identify values with the highest frequency.

Both 4 and 7 appear most frequently (3 times each)

Therefore, the modes are 4 and 7 (bimodal).

Find the mode of: 1, 2, 3, 4, 5

Step 1: Count the frequency of each value.

Each value appears exactly once.

Step 2: Identify values with the highest frequency.

All values have the same frequency, so there is no single most frequent value.

Therefore, this dataset has no mode.

For grouped data, the modal class is the class with the highest frequency. To estimate the mode within that class, we can use this formula:

Mode = L + [(d₁) / (d₁ + d₂)] × w
Where:
L = Lower boundary of the modal class
d₁ = Frequency of modal class - Frequency of previous class
d₂ = Frequency of modal class - Frequency of next class
w = Width of the modal class

Find the mode of this grouped data:

Class	Frequency
10-19	8
20-29	15
30-39	25
40-49	12
50-59	5

Step 1: Identify the modal class

Modal class = 30-39 (highest frequency = 25)

Step 2: Apply the formula

L = 30

d₁ = 25 - 15 = 10

d₂ = 25 - 12 = 13

w = 10

Mode = 30 + [(10) / (10 + 13)] × 10

= 30 + (10 / 23) × 10

= 30 + 4.35

= 34.35

Therefore, the estimated mode is 34.35.

Important Properties of Mode

The mode is the only measure of central tendency that can be used for categorical (non-numeric) data.
A dataset can have multiple modes or no mode at all.
The mode is not affected by extreme values.
The mode represents the most typical value in a dataset.

Mode Calculator

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Range (Measure of Dispersion)

The range is the simplest measure of dispersion, representing the difference between the maximum and minimum values in a dataset.

Range = Maximum value - Minimum value

Examples of Finding the Range

Find the range of: 12, 8, 20, 15, 7, 16

Step 1: Identify the maximum and minimum values.

Maximum value = 20

Minimum value = 7

Step 2: Calculate the difference.

Range = 20 - 7 = 13

Therefore, the range is 13.

Find the range of: -5, 3, -10, 8, 0, -7, 4

Step 1: Identify the maximum and minimum values.

Maximum value = 8

Minimum value = -10

Step 2: Calculate the difference.

Range = 8 - (-10) = 8 + 10 = 18

Therefore, the range is 18.

Find the range of this grouped data:

Class	Frequency
20-29	8
30-39	15
40-49	22
50-59	18
60-69	7

Step 1: Identify the lower boundary of the lowest class and the upper boundary of the highest class.

Lower boundary of lowest class = 20

Upper boundary of highest class = 69

Step 2: Calculate the difference.

Range = 69 - 20 = 49

Therefore, the range is 49.

Other Measures of Dispersion

While range is the simplest measure of dispersion, it's often used alongside other measures:

The interquartile range (IQR) measures the spread of the middle 50% of values and is less sensitive to outliers than the range.

IQR = Q₃ - Q₁
Where Q₁ is the first quartile (25th percentile) and Q₃ is the third quartile (75th percentile)

Find the IQR of: 5, 7, 10, 12, 15, 18, 20, 22, 25

Step 1: Find Q₁ (the median of the lower half).

Lower half: 5, 7, 10, 12

Q₁ = (7 + 10) / 2 = 8.5

Step 2: Find Q₃ (the median of the upper half).

Upper half: 18, 20, 22, 25

Q₃ = (20 + 22) / 2 = 21

Step 3: Calculate the IQR.

IQR = Q₃ - Q₁ = 21 - 8.5 = 12.5

The variance measures the average squared distance from the mean, providing a measure of how spread out the values are.

Population Variance (σ²) = Σ(x - μ)² / N
Sample Variance (s²) = Σ(x - x̄)² / (n-1)

Find the sample variance of: 4, 7, 9, 12, 15

Step 1: Calculate the mean.

Mean = (4 + 7 + 9 + 12 + 15) / 5 = 47 / 5 = 9.4

Step 2: Find the squared deviations from the mean.

(4 - 9.4)² = (-5.4)² = 29.16

(7 - 9.4)² = (-2.4)² = 5.76

(9 - 9.4)² = (-0.4)² = 0.16

(12 - 9.4)² = (2.6)² = 6.76

(15 - 9.4)² = (5.6)² = 31.36

Step 3: Sum the squared deviations and divide by (n-1).

Sum of squared deviations = 29.16 + 5.76 + 0.16 + 6.76 + 31.36 = 73.2

Variance = 73.2 / (5-1) = 73.2 / 4 = 18.3

The standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the original data.

Population Standard Deviation (σ) = √σ²
Sample Standard Deviation (s) = √s²

Find the sample standard deviation of: 4, 7, 9, 12, 15

Step 1: Calculate the variance (from previous example).

Variance = 18.3

Step 2: Take the square root.

Standard Deviation = √18.3 ≈ 4.28

Important Properties of Range

The range is highly sensitive to outliers.
The range only considers the two extreme values and ignores all other values in between.
The range is easy to calculate but provides limited information about the distribution.
For more detailed analysis of dispersion, other measures like IQR, variance, or standard deviation are often preferred.

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Comparison of Measures

Measure	Formula	Best Used When	Limitations
Mean	μ = Σx / n	Data is symmetrically distributed Performing further statistical analysis	Sensitive to outliers May not represent the "typical" value in skewed distributions
Median	Middle value when sorted	Data is skewed Dataset contains outliers Dealing with ordinal data	Ignores the actual values (only considers position) Less useful for further statistical analysis
Mode	Most frequent value	Categorical data Finding the most typical value Bimodal/multimodal distributions	May not exist or may not be unique Not suitable for continuous data without grouping
Range	Max - Min	Quick assessment of spread Small datasets	Extremely sensitive to outliers Only considers two values

Test Your Understanding

Statistical Measures Quiz

Question 1: Find the mean of the following dataset: 12, 15, 18, 23, 27

15 18 19 20

Question 2: Find the median of the following dataset: 7, 4, 10, 8, 5, 12

7 7.5 8 7.6

Question 3: Find the mode(s) of the following dataset: 5, 8, 5, 10, 12, 8, 5, 15

5 8 5 and 8 No mode

Question 4: Find the range of the following dataset: -5, 2, 8, -3, 10, 7, -8

15 18 10 13

Question 5: A student scored 75, 82, 78, and 89 on four tests. What score must they get on the fifth test to have a mean score of exactly 80?

70 76 80 85

Question 6: For the grouped data below, which class contains the median?

Class	Frequency	Cumulative Frequency
0-10	5	5
11-20	8	13
21-30	15	28
31-40	12	40
41-50	10	50

0-10 11-20 21-30 31-40

Question 7: Which measure of central tendency is most affected by extreme values?

Mean Median Mode All are equally affected

Question 8: Find the weighted mean if values 10, 15, and 20 have weights 2, 3, and 5 respectively.

15 16 16.5 17

Question 9: What is the interquartile range (IQR) for the dataset: 5, 8, 12, 15, 19, 22, 25, 28, 32?

14 17 20 27

Question 10: Which measure would be most appropriate for finding the central tendency of categorical data like favorite colors?

Mean Median Mode Range

Summary and Key Points

Mean

The arithmetic average of all values
Formula: Mean = Σx ÷ n
Best for symmetric distributions
Sensitive to outliers
Used for further statistical analysis

Median

The middle value when data is arranged in order
Not affected by extreme values
Best for skewed distributions
Requires different methods for odd and even counts

Mode

The most frequently occurring value
The only measure usable for categorical data
Can have multiple modes or no mode
Represents the most typical value

Range

The difference between the maximum and minimum values
Formula: Range = Max - Min
Simple measure of dispersion
Highly sensitive to outliers
Often used alongside IQR, variance, or standard deviation

Remember to choose the appropriate measure based on your data type and distribution. For a complete analysis, it's often beneficial to calculate multiple measures rather than relying on just one.