Section B — Chapters 23–39
Statistics III: Representing & Interpreting Data Numerically
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to calculate the mode, median, and mean of a set of data.
- ✓By the end of this lesson students will be able to determine the most appropriate average to use in different contexts.
- ✓By the end of this lesson students will be able to calculate the range as a measure of spread.
- ✓By the end of this lesson students will be able to identify common ways statistics can be misused or misinterpreted.
Key concepts
The mode is the value that appears most frequently in a set of data. A data set can have one mode, more than one mode (multimodal), or no mode if all values appear with the same frequency.
The median is the middle value in a set of data when the data is arranged in order from smallest to largest. If there is an odd number of data values, the median is the single middle value. If there is an even number of data values, the median is the average (mean) of the two middle values.
The mean (often called the average) is calculated by adding all the values in a data set and then dividing the sum by the total number of values in the set.
The mode, median, and mean are all types of averages. They are called measures of central tendency because they describe the 'centre' or typical value of a data set. Each average has strengths and weaknesses, making it more suitable for different types of data or situations.
The range is a measure of how spread out the data is. It is calculated by subtracting the lowest value from the highest value in a data set. A larger range indicates greater variability in the data.
Statistics can sometimes be used in misleading ways, either accidentally or on purpose. Common misuses include: using small or biased samples that don't represent the whole group; presenting data with misleading graphs (e.g., truncated axes, inconsistent scales); or 'cherry-picking' data to support a particular argument while ignoring contradictory evidence.
Key facts to remember
- 1To find the median, always arrange the data in numerical order first.
- 2The mean is calculated by summing all values and dividing by the count of values.
- 3The mode is the value that appears most often; there can be no mode or multiple modes.
- 4The range measures the spread of data: Highest value - Lowest value.
- 5The mean is sensitive to extreme values (outliers), while the median is not.
- 6The median is often a better measure of central tendency than the mean when data is skewed or contains outliers.
- 7Statistics can be misleading if samples are biased or graphs are presented in a deceptive way.
Worked examples
Example 1
A student recorded the number of hours they spent studying maths each day for a week: 2, 3, 1, 2, 4, 2, 3. Calculate the mode, median, mean, and range of this data.
Answer
Mode = 2 hours, Median = 2 hours, Mean = 2.43 hours, Range = 3 hours.
Remember to always order the data before finding the median.
Example 2
The scores of 6 students in a maths test were: 65, 72, 88, 55, 72, 90. Find the mode, median, mean, and range for these scores.
Answer
Mode = 72, Median = 72, Mean = 73.67, Range = 35.
When there's an even number of data points, the median is the average of the two middle values.
Example 3
A small company has 5 employees. Their annual salaries are: €25,000, €28,000, €30,000, €32,000, €150,000. Which average (mean or median) would best represent a 'typical' salary for an employee in this company, and why?
Answer
The median (€30,000) would best represent a 'typical' salary. This is because the mean (€53,000) is skewed upwards by the single very high salary (€150,000), which is an outlier. The median is less affected by extreme values.
Outliers can significantly distort the mean, making the median a more appropriate average in such cases.
Common mistakes
- ✗Not ordering the data before attempting to find the median.
- ✗Forgetting to divide by the total number of values when calculating the mean.
- ✗Confusing the definitions of mode, median, and mean.
- ✗Incorrectly calculating the median when there is an even number of data points (e.g., just picking one of the middle two values instead of averaging them).
- ✗Failing to identify all modes when a data set is multimodal.
Exam tips
- ★Always show your full working for calculations, especially for mean and median, as partial marks can be awarded.
- ★Read the question carefully to determine which average (mode, median, or mean) is being asked for, or if you need to decide which is most appropriate.
- ★Double-check your ordering of data for the median and your arithmetic for the mean.
- ★When asked to explain why a particular average is best, refer to the presence of outliers or the nature of the data (e.g., categorical data for mode).
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