11.2.1 Arithmetic Mean or Simply Mean

Introduction

The arithmetic mean, commonly called the mean or average, is the most frequently used measure of central tendency. It provides a single value that represents the entire dataset by balancing all values. This topic explains the definition, formula, and calculation of the mean for both ungrouped and grouped data.

Definition

Mean is the sum of all the values of a 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.

Main Idea

The arithmetic mean is a single value calculated to represent the whole data. It may be calculated for a data sample of patients or a population of migrating birds, etc. The mean expresses the central idea of the observations.

Core Concept

Formula for Mean

Where:

• x = mean
• ∑ x = sum of all values
• n = total number of values
• f = frequency
• x = midpoint of class interval (lower limit + upper limit / 2)

Detailed Explanation

1. Mean for Ungrouped Data

Definition: Mean is the sum of all the values of data set divided by the total number of values in the data set.

Formula:

x = ∑ xn

Where:

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

Simple explanation: Add up all the numbers and divide by how many numbers there are.

2. Mean for Grouped Data

For grouped data, the mean is calculated using the frequency of each class interval.

Formula:

x = ∑ fx∑ f

Where:

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

Simple explanation: Multiply the midpoint of each class by its frequency, add them up, and divide by the total frequency.

Solved Example (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.

Solved Example (Grouped Data)

The following table shows the heights of plants (in cm) and their frequencies. Calculate the mean height.

Solution:

Given:

• ∑ f = 100
• ∑ fx = 4950

Required: Mean

Formula:

x = ∑ fx∑ f

Substitution:

x = 4950100

Calculation:

x = 49.5

Final Answer: Mean height = 49.5 cm

Summary Table

Intermediate Level

The mean is the sum of all values divided by the total number of values.

Formula for ungrouped data:

x = ∑ xn

Formula for grouped data:

x = ∑ fx∑ f

Key points:

• The mean uses all values in the dataset.
• It is affected by extreme values (outliers).
• The symbol for mean is x (x bar).

Example (ungrouped): Mean of 10, 20, 30, 40, 50 = 1505 = 30

Example (grouped): Use midpoints and frequencies to calculate the mean.

Important Points

• - 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.
• - 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 of Mean for ungrouped data is: x = ∑ xn
• - Formula of Mean for grouped data is: x = ∑ fx∑ f
• - Where x = lower limit + upper limit / 2 and f = frequency distribution.

Remember

• - Mean = average
• - Symbol = x (x bar)
• - Ungrouped formula = ∑ xn
• - Grouped formula = ∑ fx∑ f
• - Mean uses all data values
• - Mean is affected by extreme values
• - Example: Mean of 2, 4, 6 = 4

Common Mistakes

• 1. Forgetting to divide by the total number of values.
• - The mean is the sum divided by the count.
• 2. Mixing up ungrouped and grouped formulas.
• - Ungrouped = ∑ xn
• - Grouped = ∑ fx∑ f
• 3. Using the wrong midpoint for grouped data.
• - Midpoint = lower limit + upper limit2
• 4. Thinking the mean is always the best measure.
• - The mean is affected by extreme values. The median may be better for skewed data.
• 5. Forgetting the symbol for mean.
• - Mean is denoted by x (x bar).
• 6. Incorrectly adding frequencies.
• - Make sure ∑ f is the total frequency.
• 7. Not checking for outliers.
• - Outliers can significantly affect the mean.

Quick Revision

60-Second Revision

• **Mean** = average
• Symbol = x (x bar)
• Ungrouped formula = ∑ xn
• Grouped formula = ∑ fx∑ f
• Mean uses all data values
• Mean is affected by extreme values
• **Example:** Mean of 2, 4, 6 = 4
• Warning: Do not forget to divide by the total number of values. Use the correct formula for grouped and ungrouped data.