11.2 Definition and Calculation of Mean, Median and Mode

Introduction

In our daily life, we commonly hear statements like "the average heartbeat is 72 beats per minute" or "the average rainfall in Islamabad is 1500 mm per year." These statements are not exactly true at every moment, but they give us a general idea. In statistics, these are called averages or measures of central tendency. This topic explains three common types of averages: mean, median, and mode.

Definition

Average: A single value which is calculated to represent the whole data. It expresses the central idea of the observations.

Types of averages commonly used:

• Arithmetic Mean
• Median
• Mode

(Also: Geometric mean, Harmonic mean)

Main Idea

In our daily conversation, we make many statements which have some meaning only on average basis. In different fields of life like agriculture, health, poultry, the idea of average is very important. Many experts at national and international level discuss the findings of studies in averages.

The average is also called measure of central tendency.

Why this is important:

• Averages help summarize large amounts of data.
• They provide a single value that represents the whole dataset.
• They are used in agriculture, health, medicine, and research.

Core Concept

Types of Averages

Detailed Explanation

1. Arithmetic Mean (or Simply Mean)

Definition: Mean is the sum of all the values of data set divided by the total number of values in the data set. It is the single value which is calculated to represent the whole set of data.

Symbol: The symbol "x" (read as "x bar") represents the sample mean. The bar above the letter x represents the mean of a set of values.

Formula:

Mean = Sum of valuesTotal number of values

For ungrouped data:

x = ∑ xn

Where:

• ∑ x = sum of all values
• n = total number of values

For grouped data:

x = ∑ fx∑ f

Where:

• x = midpoint of class interval (lower limit + upper limit / 2)
• f = frequency of each class

Simple explanation: The mean is what we usually call the "average." Add up all the numbers and divide by how many numbers there are.

2. Median

Definition: The median is the middle value in a dataset that is arranged in ascending or descending order of magnitude. Median divides the data set into two equal parts.

• One part of the dataset has values less than the middle value.
• The other part of the dataset has values greater than the middle item.

Symbol: It is denoted by x (read as "x tilde" or "x snake").

Important: It is necessary to arrange the values of dataset in ascending or descending order.

Advantages:

• The value of median is not affected by change in extreme values of dataset.
• It is easy to calculate and is located exactly.

For ungrouped data:

If the number of values is odd:

Median (x) = Value of (n+12)^th item

If the number of values is even:

Median (x) = (n2)^th value + (n2 + 1)^th value2

For grouped data:

If the number of values is odd:

Median (x) = Value of data against (n+12)^th cumulative frequency

If the number of values is even:

Median (x) = Class values against n2^th + (n2 + 1)^th cumulative frequencies2

Simple explanation: The median is the middle number when the data is arranged in order. If there are two middle numbers, take their average.

3. Mode

Definition: The mode is the value in a dataset that appears maximum number of times.

- A dataset may have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all (amodal) i.e., all values occur the same number of times.

Symbol: It may be denoted by x (read as x-hat or x cap).

Formula for ungrouped data:

Mode (x) = The most frequent or repeated value occurring in a given dataset

Formula for grouped data:

Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Where:

• l = lower boundary of modal group
• f_m = maximum frequency of a group
• f₁\) = previous frequency of \(f_m
• f₂\) = next frequency of \(f_m
• h = class interval of modal group

Simple explanation: The mode is the number that appears most often in the data.

Solved Examples

Example 1: Mean (Ungrouped Data)

Dataset: Team of World Health Organization (WHO) planned to assess the prevalence of polio disease in Pakistan. Study to record the polio cases in Pakistan continued for consecutive two years. Number of confirmed Polio patients detected each month of 2022 and 2023 are given in table below. Calculate the mean polio patients per month in each year. Also compare the prevalence of Polio disease in both years.

Solution:

Given:

• Sum of Polio cases in 2022 = 192
• Sum of Polio cases in 2023 = 144
• Total number of months = 12

Required:

• Mean polio patients per month in 2022
• Mean polio patients per month in 2023
• Comparison of prevalence

Formula:

Mean = Sum of valuesTotal number of values

Substitution:

Mean for 2022 = 19212

Mean for 2023 = 14412

Calculation:

Mean for 2022 = 16

Mean for 2023 = 12

Final Answer:

• Mean number of Polio patients per month in 2022 = 16
• Mean number of Polio patients per month in 2023 = 12
• Difference = 16 − 12 = 4

Conclusion: Prevalence of Polio disease decreased in 2023 compared to 2022 by 4 patients per month.

Example 2: Median (Ungrouped Data, Odd Number)

Dataset: The measured heights of plants after arranging in ascending order:

46, 49, 55, 58, 62, 63, 64, 65, 66, 67, 68, 70, 72

Solution:

Given:

- Number of values in the dataset = 13 (odd)

Required: Median

Formula:

Median (x) = Value of (n+12)^th item

Substitution:

Median (x) = 13+12 = 142 = 7

Calculation:

The 7th value in the dataset is 64.

Final Answer: Median = 64

Example 3: Median (Ungrouped Data, Even Number)

Dataset: The number of fruits produced on each of the 16 plants of experimental group:

3, 5, 18, 21, 15, 10, 8, 12, 13, 7, 11, 14, 9, 16, 20, 24

Solution:

Given:

- Number of values = 16 (even)

Required: Median

Step 1: Arrange in ascending order:

3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 24

Formula:

Median (x) = (n2)^th value + (n2 + 1)^th value2

Substitution:

Median (x) = (162)^th value + (162 + 1)^th value2

= 8^th value + 9^th value2

Calculation:

8th value = 12; 9th value = 13

Median (x) = 12 + 132 = 252 = 12.5

Final Answer: Median = 12.5 fruits per plant

Example 4: Mode (Ungrouped Data)

Dataset: Find the mode from the word "BIOLOGY".

Solution:

Given: The word "BIOLOGY"

Required: Mode

Formula:

Mode (x) = The most repeated value in the data

Calculation:

Letters in "BIOLOGY": B, I, O, L, O, G, Y

The letter "O" appears twice. All other letters appear once.

Final Answer: Mode = O

Example 5: Mode (Ungrouped Data)

Dataset: 3, 5, 6, 6, 9, 10, 9, 6, 6

Solution:

Given: 3, 5, 6, 6, 9, 10, 9, 6, 6

Required: Mode

Calculation:

• 3 appears 1 time
• 5 appears 1 time
• 6 appears 4 times
• 9 appears 2 times
• 10 appears 1 time

The number 6 appears most frequently (4 times).

Final Answer: Mode = 6

Example 6: Mode (Ungrouped Data, Bimodal)

Dataset: 3, 5, 5, 6, 9, 9, 10, 12

Solution:

Given: 3, 5, 5, 6, 9, 9, 10, 12

Required: Mode

Calculation:

• 3 appears 1 time
• 5 appears 2 times
• 6 appears 1 time
• 9 appears 2 times
• 10 appears 1 time
• 12 appears 1 time

Both 5 and 9 appear most frequently (2 times each).

Final Answer: Mode = 5 and 9 (bimodal)

Summary Table

Intermediate Level

Mean, median, and mode are measures of central tendency.

Formulas:

• Mean (ungrouped): x = ∑ xn
• Mean (grouped): x = ∑ fx∑ f
• Median (odd): (n+12)^th value
• Median (even): Average of n2 and n2+1 values
• Mode (ungrouped): Most repeated value
• Mode (grouped): l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Key points:

• Median is not affected by extreme values.
• Mode can be unimodal, bimodal, or multimodal.
• Mean is the most commonly used average.

Important Points

• - In our daily life routine, we commonly hear the somewhat similar to following statements.
• - Heartbeat of human is 72 beats per minute.
• - Death rate due to cancer in Pakistan is 25 per thousand patients.
• - Production of wheat in Punjab is 2000 Kg per acer.
• - The price of the bananas in the market is Rs. 120 per dozen.
• - Food consumption of a human is 1kg per day.
• - The rain fall in Islamabad is 1500mm per year.
• - If we think on about above mentioned statements, none of them is found exactly correct.
• - In terms of statistics, we call such statements as average statements.
• - In our daily conversation, we make many statements which have some meaning only on average basis.
• - In different fields of life like agriculture, health, poultry, the idea of average is very important.
• - Many experts at national and international level discuss the findings of studies in averages.
• - The average is also called measure of central tendency.
• - Average is a single value which is calculated to represent the whole data.
• - Types of averages: (i) Arithmetic Mean (ii) Median (iii) Mode (iv) Geometric mean (v) Harmonic mean
• - Mean is the sum of all the values of data set divided by the total number of values in the data set.
• - The symbol "x" (read as "x bar") represent the sample mean.
• - Formula of Mean for un-group data: x = ∑ xn
• - Formula of Mean for group data: x = ∑ fx∑ f
• - The median is the middle value in a dataset that are arranged in ascending or descending order of magnitude.
• - Median divides the data set into two equal parts.
• - It is denoted by x (read as "x tilde" or "x snake").
• - It is necessary to arrange the values of dataset in ascending or descending order.
• - The value of median is not affected by change in extreme values of dataset.
• - The mode is the value in a dataset that appears maximum number of times.
• - A dataset may have one mode (unimodal), more than one mode (bimodal or multimodal) or no mode at all (amodal) i.e. all values occur in same number of times.
• - It may be denoted by x (read as x-hat or x cap).
• - Mode = the most frequent or repeated value occurring in a given dataset.
• - Formula for calculating Mode for Group data is: Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Remember

• - Average = single value representing whole data
• - Types = mean, median, mode
• - Mean = sum of values / total number of values
• - Median = middle value when data is arranged in order
• - Mode = most frequently occurring value
• - Mean symbol = x
• - Median symbol = x
• - Mode symbol = x
• - Median is not affected by extreme values
• - Mode can be unimodal, bimodal, or multimodal

Common Mistakes

• 1. Confusing mean, median, and mode.
• - Mean = average (sum divided by count).
• - Median = middle value.
• - Mode = most frequent value.
• 2. Forgetting to arrange data in order for median.
• - Data must be arranged in ascending or descending order before finding the median.
• 3. Thinking the median is always in the dataset.
• - For even number of values, the median is the average of two middle values and may not be in the dataset.
• 4. Mixing up the formulas for grouped and ungrouped data.
• - Ungrouped: x = ∑ xn
• - Grouped: x = ∑ fx∑ f
• 5. Thinking there is always one mode.
• - A dataset can have no mode, one mode, or multiple modes.
• 6. Believing the mean is always the best measure.
• - The mean is affected by extreme values. The median may be better for skewed data.
• 7. Forgetting the symbol for mode.
• - Mode is denoted by x (x-hat).

Quick Revision

60-Second Revision

• Average = single value representing whole data
• Types = mean, median, mode
• Mean = sum of values / total number of values
• Median = middle value when data is arranged in order
• Mode = most frequently occurring value
• Mean symbol = x
• Median symbol = x
• Mode symbol = x
• Median is not affected by extreme values
• Mode can be unimodal, bimodal, or multimodal
• Warning: Do not confuse mean (average) with median (middle value) or mode (most frequent value). Always arrange data in order before finding the median.