Learning Objectives
- Calculate and interpret the mean, median, mode and range for individual data.
- Calculate averages from frequency tables.
- Estimate the mean of grouped discrete or continuous data using class midpoints.
- Identify the modal class in grouped data.
- Compare data sets using averages and measures of spread while recognising limitations.
Key Terms
- Mean
- The total of the values divided by the number of values.
- Median
- The middle value when data are ordered, or the mean of the two middle values.
- Mode
- The value occurring most frequently.
- Range
- The largest value minus the smallest value.
- Frequency
- The number of times a value or class occurs.
- Class interval
- A range used to group values.
- Class midpoint
- The value halfway between the class boundaries, used to estimate a grouped mean.
- Modal class
- The class interval with the greatest frequency.
What This Chapter Covers
- Mean, median, mode and range
- Averages from discrete frequency tables
- Estimated mean from grouped data
- Modal class and choice of average
- Comparison and interpretation of data sets
Why More Than One Average Is Needed
An average describes a typical value, but different averages emphasise different features. The mean uses every value and is sensitive to extreme values. The median depends on order and is resistant to outliers. The mode identifies the most common value and can be used with categorical data. The range is not an average, but it gives a simple measure of spread.
| Measure | How It Is Found | Strength | Limitation |
|---|---|---|---|
| Mean | Sum of values ÷ number of values | Uses every value | Affected by outliers |
| Median | Middle ordered value | Resistant to extreme values | Does not use the size of every value |
| Mode | Most frequent value | Works for categories; identifies most common | May be absent or have more than one value |
| Range | Maximum − minimum | Quick indication of spread | Uses only two values and is affected by outliers |
Mean Of Individual Data
Worked Example: Mean
Question: Find the mean of 7, 11, 12, 15, 15, 18.
- Sum =7+11+12+15+15+18=78.
- Number of values =6.
- Mean =78/6.
Answer: 13.
Median
Order the data first. For an odd number of values, the median is at position (n+1)/2. For an even number, average the values in positions n/2 and n/2+1.
Worked Example: Median And Range
Question: Find the median and range of 14, 9, 21, 11, 18, 16.
- Order: 9, 11, 14, 16, 18, 21.
- The two middle values are 14 and 16, so median =(14+16)/2=15.
- Range =21-9=12.
Answer: Median 15; range 12.
Mode
A data set can be unimodal, bimodal, multimodal or have no mode. Do not confuse mode with the largest value. The mode is the value with the highest frequency.
Mean From A Frequency Table
Multiply each value x by its frequency f, sum the products, and divide by total frequency.
Worked Example: Discrete Frequency Table
Question: The values 1, 2, 3, 4 have frequencies 3, 5, 4, 2. Find the mean.
- Calculate fx: 3, 10, 12, 8.
- \sum fx=33.
- \sum f=14.
- Mean =33/14.
Answer: \frac{33}{14}\approx2.36.
Median From A Frequency Table
Use cumulative frequency or count through the ordered frequencies until the middle position is reached. The table values are already in order only if listed from smallest to largest.
Worked Example: Median From Frequencies
Question: Scores 0, 1, 2, 3, 4 have frequencies 2, 4, 7, 5, 2. Find the median.
- Total frequency =20, so use the 10th and 11th observations.
- Cumulative frequencies are 2, 6, 13, 18, 20.
- The 10th and 11th observations are both score 2.
Answer: Median 2.
Grouped Data And The Estimated Mean
When data are grouped into intervals, exact individual values are unknown. Assume every value in a class is represented by the class midpoint. The resulting mean is therefore an estimate.
Estimated Mean Method
- Find the midpoint of each class: (\text{lower boundary}+\text{upper boundary})/2.
- Multiply each midpoint by its class frequency.
- Add the products to obtain \sum fx.
- Divide by total frequency \sum f.
- Label the answer as an estimate.
Worked Example: Grouped Continuous Data
Question: Times in minutes are grouped as 0–10, 10–20, 20–30, 30–50 with frequencies 4, 9, 5, 2. Estimate the mean.
- Midpoints are 5, 15, 25 and 40.
- \sum fx=4(5)+9(15)+5(25)+2(40)=360.
- Total frequency =20.
- Estimated mean =360/20.
Answer: 18 minutes.
Class Boundaries And Midpoints
For continuous data written as 10\le x<20, midpoint is 15. For values rounded to the nearest unit and written 10–19, the class boundaries may be 9.5 and 19.5, still giving midpoint 14.5. Read the class notation carefully.
Modal Class
The modal class is the group with the highest frequency, not necessarily the class containing the largest values. If class widths differ, a histogram uses frequency density, but the syllabus statement for averages asks identification of the modal class from a grouped frequency distribution. In an ordinary frequency table, use the largest frequency.
Choosing The Most Suitable Average
| Situation | Usually Most Useful | Reason |
|---|---|---|
| Income data with a few extremely high earners | Median | Less affected by extreme values |
| Shoe sizes to stock in a shop | Mode | Identifies the most frequently needed size |
| Repeated measurements with no serious outliers | Mean | Uses all observations |
| Open-ended grouped data | Median or mode may be safer | A reliable midpoint for an open class may not exist |
Comparing Two Data Sets
A good comparison uses one measure of centre and one measure of spread. For example, “Class A has a higher median, so its typical score is higher, while Class B has a smaller range, so its scores are more consistent.” Do not state that one data set is “better” without linking the conclusion to the context.
Worked Example: Compare Performance
Question: Class P has mean 68 and range 24. Class Q has mean 65 and range 12. Compare the classes.
- Class P has the higher mean, so its average score is higher.
- Class Q has the smaller range, so its scores are less spread out and more consistent.
- Neither statement alone proves every student in one class performed better.
Answer: P has higher average performance; Q is more consistent.
Limitations And Misleading Conclusions
Averages can hide the shape of a distribution. Two data sets can have the same mean but very different spreads. The range depends only on the two extreme values. A sample may be biased or too small. Grouped means are estimates because actual values are replaced by midpoints. State these restrictions when interpreting results.
Working Backwards From A Mean
Since \text{total}=\text{mean}\times\text{number of values}, missing values or combined means can be found by recovering totals.
Worked Example: Missing Value
Question: The mean of 8 numbers is 14. Seven of them total 91. Find the eighth number.
- Total of all 8 values =8\times14=112.
- Missing value =112-91.
Answer: 21.
Worked Example: Combined Mean
Question: Group A has 12 students with mean 16. Group B has 18 students with mean 21. Find the combined mean.
- Group A total =12(16)=192.
- Group B total =18(21)=378.
- Combined total =570; combined number =30.
- Combined mean =570/30.
Answer: 19.
Examination Guidance
- Always order raw data before finding the median.
- Use \sum fx/\sum f, not \sum f/\sum fx.
- For grouped data, show class midpoints and state that the mean is an estimate.
- When comparing data sets, comment on both centre and spread.
- Use totals rather than averaging two means directly when group sizes differ.
Common Mistakes
- Taking the mode as the largest number rather than the most frequent.
- Forgetting to average the two middle values when n is even.
- Dividing \sum fx by the number of classes instead of total frequency.
- Using a class endpoint instead of the midpoint for an estimated mean.
- Averaging two group means without weighting them by group size.
Knowledge Check And Practice
1. Find the mean of 4, 6, 9, 11, 15.
2. Find the median of 3, 10, 7, 8, 12, 15.
3. Find the mode of 2, 4, 4, 5, 7, 7, 7, 9.
4. Find the range of 18, 6, 25, 11, 20.
5. Values 2, 3, 4 have frequencies 5, 2, 3. Find the mean.
6. Class intervals 20–30 and 30–40 have midpoints what?
7. Which class is modal in frequencies 7, 12, 9, 5?
8. The mean of 5 values is 18. What is their total?
9. Why is a grouped mean only an estimate?
