Statistics

Averages and Spread

Year 10 · Year 11

✓By the end of this lesson students will be able to calculate the mean, median, mode, and range for a set of discrete data.
✓By the end of this lesson students will be able to understand the advantages and disadvantages of each average.
✓By the end of this lesson students will be able to calculate an estimate for the mean from grouped frequency tables.
✓By the end of this lesson students will be able to identify the modal class and the class containing the median for grouped data.

Key concepts

Mean

The mean is the most commonly used average. It is calculated by summing all the values in a data set and dividing by the total number of values. It takes into account every value in the data set but can be heavily influenced by extreme values (outliers).

Mean = (Sum of all values) / (Number of values)

Median

The median is the middle value of a data set when the values are arranged in ascending order. If there is an even number of data points, the median is the mean of the two middle values. The median is less affected by outliers than the mean, making it a good average for skewed data.

Position of median = (n + 1) / 2, where n is the number of data points.

Mode

The mode is the value that appears most often in a data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values appear with the same frequency. The mode is the only average that can be used for non-numerical (qualitative) data.

Range

The range is a measure of spread or dispersion, indicating the difference between the largest and smallest values in a data set. A larger range suggests greater variability in the data. It is not an average but a measure of how spread out the data is.

Range = Highest value - Lowest value

Grouped Data & Estimated Mean

When data is presented in a grouped frequency table, we do not have the individual data values. Therefore, we cannot calculate the exact mean. Instead, we estimate the mean by assuming that all values within each class interval are at its midpoint. The estimated mean is then calculated using these midpoints and their corresponding frequencies.

Estimated Mean = (Sum of (midpoint of class × frequency of class)) / (Sum of frequencies)

Key facts to remember

1The mean is calculated by summing all values and dividing by the count of values.
2The median is the middle value of an ordered data set; its position is found using (n+1)/2.
3The mode is the most frequent value in a data set.
4The range is a measure of spread: Highest value - Lowest value.
5For grouped data, the mean is an estimate calculated using class midpoints.
6The median is generally a better average than the mean if there are extreme values (outliers) in the data.
7The mode is the only average suitable for qualitative (non-numerical) data.
8Always include units in your final answers where appropriate.

Worked examples

Example 1

Find the mean, median, mode, and range for the following set of data: 5, 8, 3, 12, 5, 7, 10.

IStep 1: Order the data in ascending order: 3, 5, 5, 7, 8, 10, 12.

IIStep 2: Calculate the Mean. Sum of values = 3 + 5 + 5 + 7 + 8 + 10 + 12 = 50. Number of values = 7. Mean = 50 / 7 = 7.1428... ≈ 7.14 (to 2 decimal places).

IIIStep 3: Find the Median. There are 7 values, so the position of the median is (7 + 1) / 2 = 4th value. The 4th value in the ordered list is 7.

IVStep 4: Find the Mode. The value that appears most frequently is 5 (it appears twice).

VStep 5: Calculate the Range. Highest value = 12. Lowest value = 3. Range = 12 - 3 = 9.

Answer

Mean = 7.14 (2 d.p.), Median = 7, Mode = 5, Range = 9.

Always order the data first when finding the median or range to avoid errors.

Example 2

The number of goals scored by a football team in 20 matches is shown in the frequency table below. Calculate the mean, median, mode, and range for the number of goals scored. Goals | Frequency ------|---------- 0 | 3 1 | 5 2 | 8 3 | 3 4 | 1

IStep 1: Calculate the total number of matches (total frequency). Total frequency = 3 + 5 + 8 + 3 + 1 = 20.

IIStep 2: Calculate the Mean. Sum of (goals × frequency) = (0×3) + (1×5) + (2×8) + (3×3) + (4×1) = 0 + 5 + 16 + 9 + 4 = 34. Mean = 34 / 20 = 1.7.

IIIStep 3: Find the Mode. The highest frequency is 8, which corresponds to 2 goals. So, the mode is 2 goals.

IVStep 4: Find the Median. The position of the median is (20 + 1) / 2 = 10.5th value. We use cumulative frequency to find this: 3 (for 0 goals), 3+5=8 (for 0 or 1 goal), 8+8=16 (for 0, 1 or 2 goals). The 10.5th value falls within the '2 goals' category (as it's between the 9th and 16th value). So, the median is 2 goals.

VStep 5: Calculate the Range. Highest number of goals = 4. Lowest number of goals = 0. Range = 4 - 0 = 4 goals.

Answer

Mean = 1.7 goals, Median = 2 goals, Mode = 2 goals, Range = 4 goals.

For frequency tables, the median position tells you which data value it corresponds to, not the median value itself.

Example 3

The heights of 50 students are recorded in the grouped frequency table below. Estimate the mean height, state the modal class, and identify the class containing the median. Height (cm) | Frequency ------------|---------- 150 < h ≤ 160 | 8 160 < h ≤ 170 | 20 170 < h ≤ 180 | 15 180 < h ≤ 190 | 7

IStep 1: Calculate the midpoint for each class interval.

IIClass 150 < h ≤ 160: Midpoint = (150 + 160) / 2 = 155 cm.

IIIClass 160 < h ≤ 170: Midpoint = (160 + 170) / 2 = 165 cm.

IVClass 170 < h ≤ 180: Midpoint = (170 + 180) / 2 = 175 cm.

VClass 180 < h ≤ 190: Midpoint = (180 + 190) / 2 = 185 cm.

VIStep 2: Calculate the total frequency. Total frequency = 8 + 20 + 15 + 7 = 50.

VIIStep 3: Estimate the Mean. Sum of (midpoint × frequency) = (155×8) + (165×20) + (175×15) + (185×7) = 1240 + 3300 + 2625 + 1295 = 8460. Estimated Mean = 8460 / 50 = 169.2 cm.

VIIIStep 4: State the Modal Class. The class with the highest frequency is 160 < h ≤ 170 (frequency 20).

9Step 5: Identify the class containing the Median. The position of the median is (50 + 1) / 2 = 25.5th value. Using cumulative frequency: 8 (for 150 < h ≤ 160), 8+20=28 (for 160 < h ≤ 170). The 25.5th value falls within the 160 < h ≤ 170 class.

Answer

Estimated Mean = 169.2 cm, Modal Class = 160 < h ≤ 170, Class containing the Median = 160 < h ≤ 170.

Remember to use the midpoints for the estimated mean, not the class boundaries.

Common mistakes

✗Not ordering the data before attempting to find the median.
✗Confusing the position of the median with the median value itself, especially with frequency tables.
✗Using class boundaries instead of midpoints when calculating the estimated mean for grouped data.
✗Forgetting to divide by the total frequency when calculating the mean from a frequency table or grouped data.
✗Stating the frequency as the mode, rather than the data value that has the highest frequency.

Exam tips

★Always show your full working for calculations, as method marks are often awarded, even if the final answer is incorrect.
★When finding the median, explicitly write out the ordered list of data or cumulative frequencies to minimise errors.
★For grouped data, clearly list the midpoints you are using in your working.
★Read the question carefully to determine which average or measure of spread is required, and whether an estimate is acceptable.

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