Mathematics · Year 10 · Statistical Measures and Graphs · Spring Term

Measures of Central Tendency

Calculating and interpreting mean, median, and mode from raw data and frequency tables.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Statistics

About This Topic

Cumulative frequency and box plots are essential tools for analyzing the distribution and spread of data. Students learn to construct cumulative frequency graphs to find the median, quartiles, and interquartile range (IQR). This allows for a more sophisticated comparison of datasets than just using the mean. Box plots provide a visual summary of this data, highlighting the 'middle 50%' and identifying outliers. This topic is a cornerstone of GCSE Statistics and is vital for interpreting data in the media and social sciences.

Students must understand why the median and IQR are often more 'robust' measures than the mean and range, especially when data is skewed. This topic is particularly effective when students can use real-world data, such as house prices or athlete performance. Active learning through 'gallery walks' and 'collaborative investigations' helps students develop the critical eye needed to justify their statistical conclusions.

Key Questions

1. Compare the strengths and weaknesses of mean, median, and mode as measures of central tendency.
2. Evaluate which measure of central tendency is most appropriate for a given dataset.
3. Explain how outliers can affect different measures of central tendency.

Learning Objectives

• Calculate the mean, median, and mode for a given set of raw data.
• Construct and interpret frequency tables to find the mean, median, and mode.
• Compare the strengths and weaknesses of mean, median, and mode for different data distributions.
• Evaluate the most appropriate measure of central tendency for a given dataset, justifying the choice.
• Explain how the presence of outliers impacts the mean, median, and mode.

Before You Start

Basic Data Handling

Why: Students need to be able to list, order, and count data points before calculating measures of central tendency.

Averages

Why: Prior exposure to the concept of 'average' as a single value representing a set of data is foundational.

Key Vocabulary

Mean	The average of a dataset, calculated by summing all values and dividing by the number of values.
Median	The middle value in a dataset when the data is ordered from least to greatest. If there is an even number of values, it is the average of the two middle values.
Mode	The value that appears most frequently in a dataset. A dataset can have one mode, more than one mode, or no mode.
Frequency Table	A table that displays the frequency of various outcomes in a sample. Each entry shows the frequency (count) for a particular category or value.
Outlier	A data point that differs significantly from other observations in a dataset. Outliers can distort statistical measures.

Watch Out for These Misconceptions

Common Misconception

Plotting cumulative frequency at the midpoint of the class interval.

What to Teach Instead

Students often confuse this with frequency polygons. Using a 'Collaborative Investigation' helps them realise that cumulative frequency represents the total 'up to' a point, so it must be plotted at the upper class boundary.

Common Misconception

Thinking a longer 'whisker' on a box plot means there are more people in that section.

What to Teach Instead

Students often equate area/length with frequency. Peer discussion helps them understand that each section of a box plot always represents exactly 25% of the data; a longer whisker just means that 25% is more spread out.

Active Learning Ideas

Inquiry Circle: Comparing Datasets

Groups are given two sets of data (e.g., test scores from two different classes). They must calculate the cumulative frequencies, draw the graphs, and create box plots to decide which class performed 'better' and why.

45 min·Small Groups

Gallery Walk: Spot the Outlier

Box plots are displayed around the room, some with deliberate errors or extreme outliers. Students move in pairs to identify the outliers using the 1.5 x IQR rule and discuss whether they should be kept or removed from the data.

25 min·Pairs

Think-Pair-Share: Skewness and Spread

Students are shown three different box plots representing different distributions. They must individually describe the 'story' the data tells (e.g., 'most people scored high but a few did very poorly') before sharing their interpretation with a partner.

15 min·Pairs

Real-World Connections

• Sports statisticians use measures of central tendency to analyze player performance. For example, a coach might look at the median number of points scored by a basketball player over a season to understand their typical scoring output, especially if there are a few unusually high or low scoring games (outliers).
• Economists use these measures to understand income distribution. Calculating the mean income can be misleading if a few very high earners skew the average; the median income often provides a more representative picture of typical earnings for a population.
• Market researchers analyze customer survey data using central tendency. For instance, when assessing customer satisfaction ratings on a scale, the mode might indicate the most common opinion, while the median shows the midpoint of all responses.

Assessment Ideas

Quick Check

Present students with two small datasets: one with a clear outlier, and one without. Ask them to calculate the mean, median, and mode for both. Then, ask: 'Which measure best represents the 'typical' value in each dataset, and why?'

Exit Ticket

Provide students with a simple frequency table showing the number of hours students spent on homework last week. Ask them to: 1. Calculate the mean, median, and mode. 2. State which measure they think is most appropriate to describe the typical homework time and briefly justify their answer.

Discussion Prompt

Pose the question: 'Imagine you are analyzing the salaries of employees at a company. Would you prefer to report the mean salary or the median salary? Explain your reasoning, considering the potential impact of a CEO's very high salary.'

Frequently Asked Questions

What is the interquartile range (IQR) and why is it useful?

The IQR is the difference between the upper and lower quartiles. It measures the spread of the middle 50% of the data, making it a better measure of consistency than the range because it ignores extreme outliers.

How do you find the median from a cumulative frequency graph?

Find the halfway point on the vertical (cumulative frequency) axis, draw a horizontal line to the curve, and then drop a vertical line down to the horizontal axis to read the value.

How can active learning help students understand box plots?

Box plots are abstract summaries. Active learning, such as 'Collaborative Investigations' where students create their own plots from raw data, helps them see exactly where the lines come from. Comparing different plots in a 'Gallery Walk' also forces students to use comparative language (e.g., 'Class A has a higher median but Class B is more consistent'), which is exactly what is required in GCSE exam questions.

Why is the median better than the mean for skewed data?

The mean is pulled toward extreme values (outliers), which can give a misleading 'average'. The median is the middle value, so it provides a more accurate representation of what a 'typical' data point looks like in a skewed set.