Measures of Central Tendency and Spread

Averages and Spread for Discrete Data

Every time you check the average test score for your class, you are using a measure of central tendency. These measures find the "typical" value in a dataset, while measures of spread tell us how stretched out the data is.

Mean: The sum of all values divided by the total number of values ( $n$ ). It uses every piece of data.
Median: The middle value when data is ordered. Found at the position $\frac{n + 1}{2}$ .
Mode (or Modal Value): The most frequent value.
Range: The difference between the maximum and minimum values (a measure of spread).

Let's calculate these for Discrete Data (data that takes specific, separate values).

Worked Example: Mean and Median

Calculate the mean and median for the values: 5, 8, 3.

Step 1: Order the data. Ordered list: 3, 5, 8

Step 2: Calculate the Mean.

\text{Mean} = \frac{\sum x}{n}

\text{Mean} = \frac{5 + 8 + 3}{3} = \frac{16}{3} = 5.33

Step 3: Find the Median Position.

\text{Position} = \frac{n + 1}{2} = \frac{3 + 1}{2} = 2

The median is the 2nd value in the ordered list, which is 5.

Worked Example: Mode (Modal Value)

Identify the mode for the values: 4, 7, 2, 7, 9.

Step 1: Find the frequency of each value. The number 7 appears twice. All other numbers appear only once.

Step 2: Identify the most frequent value.

\text{Mode} = 7

Worked Example: Range

Calculate the range for the values: -5, 2, 8.

Step 1: Use the range formula.

\text{Range} = \text{Maximum} - \text{Minimum}

\text{Range} = 8 - (-5) = 13

Grouped Frequency Tables

When dealing with hundreds of measurements, listing every single number is impossible. Instead, we group data into a Class Interval, which gives us a summary but loses the exact original values. Because the exact values are unknown, we calculate an Estimated Mean using the Midpoint ( $x$ ) of each group. We also identify the Modal Class (the group with the highest frequency) and the Median Class (the group containing the median position).

Worked Example: Estimated Mean and Modal Class

Calculate the estimated mean, modal class, and median class for the heights of 20 plants.

Height ( $h$ , cm)	Frequency ( $f$ )	Midpoint ( $x$ )	$f \times x$	Cumulative Frequency ( $cf$ )
$0 \le h < 10$	2	5	10	2
$10 \le h < 20$	7	15	105	9
$20 \le h < 30$	8	25	200	17
$30 \le h < 40$	3	35	105	20
Total	$\sum f = 20$	-	$\sum fx = 420$	-

Step 1: Calculate the Estimated Mean.

\bar{x} = \frac{\sum fx}{\sum f}

\bar{x} = \frac{420}{20} = 21 \text{ cm}

Step 2: Find the Modal Class. The highest frequency is 8, so the modal class is $20 \le h < 30$ .

Step 3: Find the Median Class. Calculate the median position using Cumulative Frequency ( $cf$ ).

\text{Position} = \frac{n + 1}{2} = \frac{20 + 1}{2} = 10.5

Looking at the $cf$ column, the 9th value ends in the previous interval, so the 10.5th value falls into the $20 \le h < 30$ interval.

Identifying Outliers and Anomalies

Have you ever noticed one wildly different result in a science experiment? That odd value out is an Outlier (or Anomaly). Outliers are extreme data values that do not fit the general pattern. They can heavily distort the mean and range, making them less representative, though the median and Interquartile Range (IQR) remain resistant. Before analysing data, you may need to practice Cleaning Data by removing these anomalies.

Worked Example: Using the $1.5 \times \text{IQR}$ Rule

Identify any mathematical outliers if the Lower Quartile ( $Q_1$ ) $= 10$ , Upper Quartile ( $Q_3$ ) $= 20$ , and Interquartile Range (IQR) $= 10$ .

Step 1: Find the outlier boundary value.

1.5 \times \text{IQR} = 1.5 \times 10 = 15

Step 2: Calculate the Lower Boundary.

\text{Lower Boundary} = Q_1 - 15 = 10 - 15 = -5

Step 3: Calculate the Upper Boundary.

\text{Upper Boundary} = Q_3 + 15 = 20 + 15 = 35

Any data value below -5 or above 35 is considered a mathematical outlier.

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
When estimating the mean, students often divide the total $\sum fx$ by the number of rows in the table. You MUST divide by the total frequency $\sum f$ .
2
In AQA exams, examiners look for the actual subtraction step in range calculations (e.g., $15 - 3 = 12$ ). Do not just write down the final answer.
3
If a dataset has no repeating values, you must explicitly state 'No mode'. Never write '0', as examiners will mark this incorrect assuming 0 is a data value.
4
When asked to state the modal or median class, you must copy the inequality signs exactly as they appear in the table (e.g., $20 \le h < 30$ ). Never give the frequency itself as your answer.
5
When AQA asks you to compare two datasets, you must make two distinct statements: one comparing the average (using mean or median) and one comparing the spread (using range or IQR).

Key Terms(18)

Mean: The sum of all values divided by the total number of values.
Median: The middle value when all data points are arranged in ascending or descending order.
Mode: The value(s) with the highest frequency in a data set.
Modal Value: The AQA-specific term for the mode; the value that occurs most often.
Range: The numerical difference between the maximum (highest) and minimum (lowest) values.
Discrete Data: Data that can only take specific, separate values, such as shoe sizes or number of children.
Class Interval: The range of values for a specific group in a grouped frequency table.
Estimated Mean: An average calculated by assuming all data points in a grouped frequency table are equal to the class midpoint.
Midpoint: The exact middle value of a class interval, found by adding the lower and upper limits and dividing by two.
Modal Class: The class interval with the greatest frequency in a grouped data table.
Median Class: The class interval containing the middle value of the dataset.
Cumulative Frequency: A running total of frequencies used to locate positions of data points like the median.
Outlier: An extreme data value that does not fit the general pattern and lies an abnormal distance from other values.
Anomaly: A value judged not to be part of the typical variation, often caused by a recording error.
Cleaning Data: The process of removing anomalies or outliers before analysis to prevent misleading results.
Interquartile Range (IQR): A measure of statistical spread representing the difference between the upper quartile and the lower quartile ( $Q_3 - Q_1$ ).
Lower Quartile (Q₁): The value positioned one-quarter of the way through an ordered dataset.
Upper Quartile (Q₃): The value positioned three-quarters of the way through an ordered dataset.

Previous Subtopic: Histograms and Cumulative FrequencyHistograms and Cumulative Frequency Next NoteQuartiles, Box Plots and Comparing Distributions

Back to Distributions and Measures

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(18)

Mean: The sum of all values divided by the total number of values.
Median: The middle value when all data points are arranged in ascending or descending order.
Mode: The value(s) with the highest frequency in a data set.
Modal Value: The AQA-specific term for the mode; the value that occurs most often.
Range: The numerical difference between the maximum (highest) and minimum (lowest) values.
Discrete Data: Data that can only take specific, separate values, such as shoe sizes or number of children.
Class Interval: The range of values for a specific group in a grouped frequency table.
Estimated Mean: An average calculated by assuming all data points in a grouped frequency table are equal to the class midpoint.
Midpoint: The exact middle value of a class interval, found by adding the lower and upper limits and dividing by two.
Modal Class: The class interval with the greatest frequency in a grouped data table.
Median Class: The class interval containing the middle value of the dataset.
Cumulative Frequency: A running total of frequencies used to locate positions of data points like the median.
Outlier: An extreme data value that does not fit the general pattern and lies an abnormal distance from other values.
Anomaly: A value judged not to be part of the typical variation, often caused by a recording error.
Cleaning Data: The process of removing anomalies or outliers before analysis to prevent misleading results.
Interquartile Range (IQR): A measure of statistical spread representing the difference between the upper quartile and the lower quartile ( $Q_3 - Q_1$ ).
Lower Quartile (Q₁): The value positioned one-quarter of the way through an ordered dataset.
Upper Quartile (Q₃): The value positioned three-quarters of the way through an ordered dataset.

Measures of Central Tendency and Spread

Averages and Spread for Discrete Data

Mean: The sum of all values divided by the total number of values ( $n$ ). It uses every piece of data.
Median: The middle value when data is ordered. Found at the position $\frac{n + 1}{2}$ .
Mode (or Modal Value): The most frequent value.
Range: The difference between the maximum and minimum values (a measure of spread).

Let's calculate these for Discrete Data (data that takes specific, separate values).

Worked Example: Mean and Median

Calculate the mean and median for the values: 5, 8, 3.

Step 1: Order the data. Ordered list: 3, 5, 8

Step 2: Calculate the Mean.

\text{Mean} = \frac{\sum x}{n}

\text{Mean} = \frac{5 + 8 + 3}{3} = \frac{16}{3} = 5.33

Step 3: Find the Median Position.

\text{Position} = \frac{n + 1}{2} = \frac{3 + 1}{2} = 2

The median is the 2nd value in the ordered list, which is 5.

Worked Example: Mode (Modal Value)

Identify the mode for the values: 4, 7, 2, 7, 9.

Step 1: Find the frequency of each value. The number 7 appears twice. All other numbers appear only once.

Step 2: Identify the most frequent value.

\text{Mode} = 7

Worked Example: Range

Calculate the range for the values: -5, 2, 8.

Step 1: Use the range formula.

\text{Range} = \text{Maximum} - \text{Minimum}

\text{Range} = 8 - (-5) = 13

Grouped Frequency Tables

Worked Example: Estimated Mean and Modal Class

Calculate the estimated mean, modal class, and median class for the heights of 20 plants.

Height ( $h$ , cm)	Frequency ( $f$ )	Midpoint ( $x$ )	$f \times x$	Cumulative Frequency ( $cf$ )
$0 \le h < 10$	2	5	10	2
$10 \le h < 20$	7	15	105	9
$20 \le h < 30$	8	25	200	17
$30 \le h < 40$	3	35	105	20
Total	$\sum f = 20$	-	$\sum fx = 420$	-

Step 1: Calculate the Estimated Mean.

\bar{x} = \frac{\sum fx}{\sum f}

\bar{x} = \frac{420}{20} = 21 \text{ cm}

Step 2: Find the Modal Class. The highest frequency is 8, so the modal class is $20 \le h < 30$ .

Step 3: Find the Median Class. Calculate the median position using Cumulative Frequency ( $cf$ ).

\text{Position} = \frac{n + 1}{2} = \frac{20 + 1}{2} = 10.5

Looking at the $cf$ column, the 9th value ends in the previous interval, so the 10.5th value falls into the $20 \le h < 30$ interval.

Identifying Outliers and Anomalies

Worked Example: Using the $1.5 \times \text{IQR}$ Rule

Identify any mathematical outliers if the Lower Quartile ( $Q_1$ ) $= 10$ , Upper Quartile ( $Q_3$ ) $= 20$ , and Interquartile Range (IQR) $= 10$ .

Step 1: Find the outlier boundary value.

1.5 \times \text{IQR} = 1.5 \times 10 = 15

Step 2: Calculate the Lower Boundary.

\text{Lower Boundary} = Q_1 - 15 = 10 - 15 = -5

Step 3: Calculate the Upper Boundary.

\text{Upper Boundary} = Q_3 + 15 = 20 + 15 = 35

Any data value below -5 or above 35 is considered a mathematical outlier.

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
When estimating the mean, students often divide the total $\sum fx$ by the number of rows in the table. You MUST divide by the total frequency $\sum f$ .
2
In AQA exams, examiners look for the actual subtraction step in range calculations (e.g., $15 - 3 = 12$ ). Do not just write down the final answer.
3
If a dataset has no repeating values, you must explicitly state 'No mode'. Never write '0', as examiners will mark this incorrect assuming 0 is a data value.
4
When asked to state the modal or median class, you must copy the inequality signs exactly as they appear in the table (e.g., $20 \le h < 30$ ). Never give the frequency itself as your answer.
5
When AQA asks you to compare two datasets, you must make two distinct statements: one comparing the average (using mean or median) and one comparing the spread (using range or IQR).

Key Terms(18)

Mean: The sum of all values divided by the total number of values.
Median: The middle value when all data points are arranged in ascending or descending order.
Mode: The value(s) with the highest frequency in a data set.
Modal Value: The AQA-specific term for the mode; the value that occurs most often.
Range: The numerical difference between the maximum (highest) and minimum (lowest) values.
Discrete Data: Data that can only take specific, separate values, such as shoe sizes or number of children.
Class Interval: The range of values for a specific group in a grouped frequency table.
Estimated Mean: An average calculated by assuming all data points in a grouped frequency table are equal to the class midpoint.
Midpoint: The exact middle value of a class interval, found by adding the lower and upper limits and dividing by two.
Modal Class: The class interval with the greatest frequency in a grouped data table.
Median Class: The class interval containing the middle value of the dataset.
Cumulative Frequency: A running total of frequencies used to locate positions of data points like the median.
Outlier: An extreme data value that does not fit the general pattern and lies an abnormal distance from other values.
Anomaly: A value judged not to be part of the typical variation, often caused by a recording error.
Cleaning Data: The process of removing anomalies or outliers before analysis to prevent misleading results.
Interquartile Range (IQR): A measure of statistical spread representing the difference between the upper quartile and the lower quartile ( $Q_3 - Q_1$ ).
Lower Quartile (Q₁): The value positioned one-quarter of the way through an ordered dataset.
Upper Quartile (Q₃): The value positioned three-quarters of the way through an ordered dataset.

Previous Subtopic: Histograms and Cumulative FrequencyHistograms and Cumulative Frequency Next NoteQuartiles, Box Plots and Comparing Distributions

Back to Distributions and Measures

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(18)

Mean: The sum of all values divided by the total number of values.
Median: The middle value when all data points are arranged in ascending or descending order.
Mode: The value(s) with the highest frequency in a data set.
Modal Value: The AQA-specific term for the mode; the value that occurs most often.
Range: The numerical difference between the maximum (highest) and minimum (lowest) values.
Discrete Data: Data that can only take specific, separate values, such as shoe sizes or number of children.
Class Interval: The range of values for a specific group in a grouped frequency table.
Estimated Mean: An average calculated by assuming all data points in a grouped frequency table are equal to the class midpoint.
Midpoint: The exact middle value of a class interval, found by adding the lower and upper limits and dividing by two.
Modal Class: The class interval with the greatest frequency in a grouped data table.
Median Class: The class interval containing the middle value of the dataset.
Cumulative Frequency: A running total of frequencies used to locate positions of data points like the median.
Outlier: An extreme data value that does not fit the general pattern and lies an abnormal distance from other values.
Anomaly: A value judged not to be part of the typical variation, often caused by a recording error.
Cleaning Data: The process of removing anomalies or outliers before analysis to prevent misleading results.
Interquartile Range (IQR): A measure of statistical spread representing the difference between the upper quartile and the lower quartile ( $Q_3 - Q_1$ ).
Lower Quartile (Q₁): The value positioned one-quarter of the way through an ordered dataset.
Upper Quartile (Q₃): The value positioned three-quarters of the way through an ordered dataset.