Mathematics · Secondary 2 · Data Handling and Probability · Semester 2

Measures of Central Tendency: Median and Mode

Calculating and comparing median and mode for various data sets, including grouped data.

• MOE Syllabus Outcomes
• MOE: Data Analysis - S2

About This Topic

Measures of central tendency summarize data by highlighting typical values, and Secondary 2 students focus on median and mode alongside the mean. The median is the middle value in an ordered list, resistant to outliers like high executive salaries, which explains why businesses prefer it for wage reports. The mode identifies the most frequent value, useful for sales data on popular sizes. Students calculate these for raw and grouped data sets, interpreting results in context.

This topic fits within the Data Handling and Probability unit, building statistical reasoning skills aligned with MOE standards. Learners compare measures to select the best for skewed distributions or multimodal data, addressing key questions on representation and interpretation. Practice reinforces when mode suits categorical data, median handles extremes, and mean averages all values.

Active learning benefits this topic greatly. Students engage deeply when collecting real data, such as class heights or snack preferences, then compute measures collaboratively. Physical sorting of data cards or tallying frequencies makes abstract ideas concrete, reveals data shapes visually, and sparks discussions on real-world choices.

Key Questions

1. Why might a business prefer to use the median rather than the mean to describe salaries?
2. Under what circumstances is the mode the most representative value of a data set?
3. Differentiate between mean, median, and mode in terms of their calculation and interpretation.

Learning Objectives

• Calculate the median for both ungrouped and grouped data sets.
• Determine the mode for various data sets, including identifying multiple modes or no mode.
• Compare the median and mode to the mean, explaining which measure is most appropriate for skewed data.
• Interpret the meaning of the median and mode within the context of a given data set.
• Differentiate between the calculation and interpretation of mean, median, and mode.

Before You Start

Measures of Central Tendency: Mean

Why: Students need to understand how to calculate and interpret the mean before comparing it with the median and mode.

Data Organization and Frequency Tables

Why: Students must be able to organize raw data and understand frequency tables to calculate median and mode, especially for grouped data.

Key Vocabulary

Median	The middle value in a data set when the data is arranged in ascending or descending order. If there is an even number of data points, it is the average of the two middle values.
Mode	The value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode.
Grouped Data	Data that has been organized into categories or intervals, often presented in a frequency table. Calculations for median and mode require specific methods for this type of data.
Outlier	A data point that is significantly different from other observations in the data set. The median is less affected by outliers than the mean.

Watch Out for These Misconceptions

Common Misconception

The median is the average of the two middle values only for even data sets.

What to Teach Instead

Median is always the middle position; for even counts, average the two central values after ordering. Active sorting with cards lets students physically find positions, clarifying rules through trial with varied set sizes.

Common Misconception

Mode works only for whole numbers, not grouped continuous data.

What to Teach Instead

Mode is the most frequent value or modal class interval in grouped data. Hands-on tallying and histogram building in groups helps students spot peaks visually, distinguishing discrete from continuous cases.

Common Misconception

All measures give the same typical value for any data.

What to Teach Instead

Choices depend on distribution; mean skews with outliers, median centers, mode peaks. Comparing computed values from shared data sets in discussions reveals context-specific strengths.

Active Learning Ideas

Data Card Sort: Median and Mode Hunt

Provide sets of printed data cards like test scores or shoe sizes. In pairs, students order cards to find median, tally for mode, and note differences from mean. Discuss why median resists one outlier score.

30 min·Pairs

Survey Station: Class Favorites

Conduct a quick whole-class survey on topics like favorite sports. Tally responses, calculate mode, and order for median age of fans. Groups present findings and compare to mean.

45 min·Small Groups

Grouped Data Challenge: Frequency Tables

Give printed frequency tables for grouped heights. Students estimate median from cumulative frequencies and identify mode interval. Pairs verify with dot plots drawn on mini-whiteboards.

35 min·Pairs

Real-World Skew: Salary Scenarios

Distribute salary data sets, one skewed. Individuals calculate all measures, then share in small groups why median best represents typical pay. Vote on business choice.

40 min·Small Groups

Real-World Connections

• Retail managers use the mode to identify the most popular sizes of clothing or shoes to stock, ensuring they have sufficient inventory of commonly purchased items.
• Human resources departments often report the median salary for a position rather than the mean, as a few very high executive salaries can skew the average, making the median a more accurate representation of typical earnings.
• Market researchers use the mode to find the most frequent response in surveys, such as the most popular color of a new car model or the most common age group for a product.

Assessment Ideas

Exit Ticket

Provide students with a small data set (e.g., test scores). Ask them to calculate the median and mode. On the back, have them write one sentence explaining which measure better represents a 'typical' score and why.

Quick Check

Present a frequency table of grouped data. Ask students to identify the modal class. Then, ask them to explain the steps they would take to find the median of this grouped data, without performing the full calculation.

Discussion Prompt

Pose the scenario: 'A company reports that the median income for its employees is $50,000, while the mean income is $75,000.' Ask students to explain why these two values are different and what this tells them about the company's salary distribution.

Frequently Asked Questions

Why use median instead of mean for salaries in business?

Salaries often skew due to a few high earners, pulling the mean upward. Median, as the middle value, better represents typical pay for most employees. Students grasp this by plotting skewed data and overlaying measures, seeing visually how outliers distort mean but not median.

When is the mode the most representative measure?

Mode shines for categorical or multimodal data, like most common jersey size in a shop or peak test score range. It highlights frequencies ignored by mean or median. Class surveys followed by tally charts make this intuitive, as students identify real 'favorites' in their data.

How to calculate median and mode for grouped data?

For grouped data, median uses cumulative frequency to find the middle position's class, interpolating if needed. Mode is the modal class with highest frequency. Practice with printed tables and graphing tools builds accuracy; peer checks ensure correct cumulative builds.

How can active learning help students understand median and mode?

Active methods like data collection and physical sorting engage students kinesthetically. Pairs ordering cards or groups building tallies reveal patterns hands-on, outperforming worksheets. Discussions on real contexts, such as school survey results, connect calculations to decisions, boosting retention and critical choice of measures.