Measurement, Space & Statistics
Comparing Data and Relative Frequency
Year 9
- ✓By the end of this lesson students will be able to calculate relative frequency from given data.
- ✓By the end of this lesson students will be able to compare two data sets using measures of centre (mean, median) and spread (range, interquartile range).
- ✓By the end of this lesson students will be able to interpret and draw conclusions from comparisons of data sets.
- ✓By the end of this lesson students will be able to understand the relationship between relative frequency and probability.
Key concepts
Relative frequency is the proportion of times a particular event or outcome occurs within a data set. It is calculated by dividing the frequency of that outcome by the total number of observations. Relative frequency provides an estimate of the probability of an event occurring based on observed data.
When comparing two or more data sets, we analyse their similarities and differences to draw meaningful conclusions. This typically involves comparing measures of centre (such as the mean or median) to understand the typical value, and measures of spread (such as the range or interquartile range) to understand the variability or consistency of the data. Visual displays like back-to-back stem-and-leaf plots and box plots are powerful tools for visual comparison.
Key facts to remember
- 1Relative frequency is calculated as (Frequency of outcome) / (Total number of observations).
- 2Relative frequency provides an estimate of the probability of an event.
- 3Measures of centre (mean, median) describe the typical or average value of a data set.
- 4Measures of spread (range, interquartile range) describe the variability or consistency within a data set.
- 5The mean is affected by outliers, while the median is more resistant to them.
- 6The range is the difference between the highest and lowest values.
- 7The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1), representing the spread of the middle 50% of the data.
- 8Visual displays like back-to-back stem-and-leaf plots and box plots are effective for comparing data sets.
Worked examples
Example 1
A survey asked 80 students about their favourite colour. The results are shown in the table below. Calculate the relative frequency for each colour, expressing your answers as decimals rounded to two decimal places.
Answer
Red: 0.30, Blue: 0.25, Green: 0.20, Yellow: 0.15, Purple: 0.10
Relative frequencies can also be expressed as fractions or percentages.
Example 2
Two Year 9 maths classes, Class A and Class B, took the same test. Their scores (out of 30) are listed below:\nClass A: 18, 22, 25, 15, 20, 23, 19, 26, 17, 21\nClass B: 16, 20, 21, 22, 24, 25, 19, 23, 18, 27\nCompare the performance of the two classes using the mean and range.
Answer
Class A: Mean = 20.6, Range = 11. Class B: Mean = 21.5, Range = 11. Conclusion: Class B had a slightly higher average score, while both classes showed similar variability in scores.
Always state both the calculated measures and a clear comparative statement.
Example 3
The daily maximum temperatures (in °C) for two cities over 9 days in autumn are:\nCity P: 18, 20, 21, 22, 23, 24, 25, 26, 28\nCity Q: 17, 19, 20, 21, 23, 25, 26, 27, 29\nCompare the temperatures of the two cities using the median and interquartile range (IQR).
Answer
City P: Median = 23°C, IQR = 5°C. City Q: Median = 23°C, IQR = 7°C. Conclusion: Both cities have the same median temperature, but City Q has a greater spread in its middle 50% of temperatures.
When calculating Q1 and Q3 for an odd number of data points, the median itself is not included in either the lower or upper half.
Common mistakes
- ✗Confusing frequency with relative frequency; relative frequency is a proportion, not a count.
- ✗Forgetting to divide by the total number of observations when calculating relative frequency.
- ✗Only calculating one measure (e.g., just the mean) when asked to compare data sets, neglecting measures of spread.
- ✗Incorrectly calculating the interquartile range (IQR), especially when dealing with an odd number of data points or misidentifying Q1 and Q3.
- ✗Failing to provide a clear, comparative statement that interprets the calculated statistics in the context of the problem.
Exam tips
- ★Always show full working for calculations of mean, median, range, and IQR to earn full marks, even if the final answer is incorrect.
- ★When comparing data sets, explicitly state what each calculated measure (e.g., 'the mean indicates...', 'the IQR shows...') tells you about the data.
- ★Read the question carefully to determine which measures of centre and spread are most appropriate for the data (e.g., median and IQR are often better for skewed data or data with outliers).
- ★Ensure your conclusions are well-supported by the statistics you have calculated and directly address the question asked.
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