๐ก For an even number of values: median = average of the two middle values.
Part 2: Range โ Measuring Spread
The range tells us how spread out the data is.
๐ Formula: Range = Highest value โ Lowest value
Example: data = 3, 7, 7, 9, 12, 15, 15, 15, 20 โ Range = 20 โ 3 = 17
๐ก A large range means the data is widely spread. A small range means the data is clustered together โ more consistent.
When comparing two groups, you must comment on both the average AND the spread. Commenting on only one gets half marks!
Part 3: Comparing Two Datasets
A complete comparison requires two statements:
1. Compare an average (mean or median) โ who scored higher/lower and by how much?
2. Compare the spread (range) โ whose results were more consistent/variable?
Template:
"Class A has a higher mean of [xฬ = 14] compared to Class B [xฬ = 9], suggesting Class A performed better on average.
Class A also has a smaller range [10] than Class B [18], so Class A's results were more consistent."
๐ก Always state the actual values in brackets โ this is what earns marks in exams.
Part 4: Back-to-Back Stem-and-Leaf Diagrams
A back-to-back stem-and-leaf diagram places two datasets either side of a shared stem, making comparison easy.
Dataset A (Class A scores): 12, 15, 23, 25, 28, 31, 34
Dataset B (Class B scores): 11, 18, 21, 22, 29, 30, 36
Class A (read right to left)
Stem
Class B (read left to right)
5
2
1
1
8
8
5
3
2
1
2
9
4
1
3
0
6
Key: 2 | 1 means 12 (Class A reads right-to-left from stem)
Median of Class A (7 values) = 4th value = 25
Median of Class B (7 values) = 4th value = 22
๐ก Always write a key! e.g. "Key: for Class A, 5|2 means 25"
Part 5: Outliers and Skew
Outlier: a value that is much higher or much lower than the rest of the data.
Dataset without outlier: 8, 9, 10, 11, 12 โ Mean = 10, Median = 10
Dataset with outlier: 8, 9, 10, 11, 12, 50 โ Mean = 16.7, Median = 10.5
๐ก The mean is pulled towards the outlier. The median barely changes. When data has outliers, the median is a more representative average.
Skew describes the shape of a distribution:
Positive Skew
Long tail to the right. Mean > Median > Mode.
Symmetric
Balanced about centre. Mean = Median = Mode.
Negative Skew
Long tail to the left. Mean < Median < Mode.
๐ก Worked Examples
Example 1: Calculating All Four Measures
Find the mean, median, mode and range for: 4, 7, 3, 9, 7, 12, 7, 5
Step 3: Median = average of 4th and 5th values = (7+7) รท 2 = 7
Step 4: Mode = most frequent value = 7 (appears 3 times)
Step 5: Range = 12 โ 3 = 9
๐ก Always order the data first! It makes finding the median and range much easier.
Example 2: Writing a Full Comparison Statement
Class A test scores: 8, 12, 14, 15, 16 Class B test scores: 5, 6, 8, 9, 12
Class A: Mean = 65รท5 = 13, Median = 14, Range = 16โ8 = 8
Class B: Mean = 40รท5 = 8, Median = 8, Range = 12โ5 = 7
Comparison (average): Class A has a higher mean (xฬ = 13) than Class B (xฬ = 8), suggesting Class A performed better overall, scoring 5 marks higher on average.
Comparison (spread): Class A has a slightly larger range (8) than Class B (7), meaning Class B's results were marginally more consistent.
โ Always compare both an average and the spread. State the actual values in your comparison.
Example 3: Reading a Back-to-Back Stem-and-Leaf
The diagram shows reaction times (ms) for Group X and Group Y.
Group X
Stem
Group Y
9
4
2
2
1
5
8
8
5
3
3
0
4
7
6
2
4
1
9
Key: For Group X, 5|3 means 35ms. For Group Y, 3|0 means 30ms.
Group X values: 22, 24, 29, 33, 35, 38, 42, 46 โ Median = (35+38)รท2 = 36.5 ms
Group Y values: 21, 25, 28, 30, 34, 37, 41, 49 โ Median = (30+34)รท2 = 32 ms
Comparison: Group X has a higher median reaction time (36.5 ms) than Group Y (32 ms), suggesting Group Y reacted faster on average.
Example 4: Outlier Effect on the Mean
A teacher records these homework scores: 6, 7, 7, 8, 8, 9, 42
Mean with outlier: (6+7+7+8+8+9+42) รท 7 = 87 รท 7 โ 12.4
Median with outlier: ordered data โ 4th value = 8
If we remove the outlier (42): Mean = 45 รท 6 = 7.5, Median = (7+8)รท2 = 7.5
Comment: The value 42 is an outlier. It pulls the mean from 7.5 to 12.4, making it an unrepresentative average. The median (8) is a better measure here as it is resistant to the outlier.
๐ก The mean is sensitive to outliers. The median is resistant (robust) to them.
๐ Interactive Visualizer
Live Dataset Comparator
Enter comma-separated numbers for each class, then press Compare!
Class A
Class B
Outlier Impact Demo
Base dataset: 5, 6, 7, 7, 8, 9, 8, 6. Watch what happens when you add an outlier!
Mean
6.875
Median
7
Back-to-Back Stem-and-Leaf Builder
Enter two datasets of two-digit numbers (10โ99). The diagram will be built automatically!
Group A
Group B
Write the Comparison Scaffold
Stats are shown below. Build your comparison sentence by selecting the correct options!
Class A: Mean = 14, Median = 13, Range = 12 |
Class B: Mean = 9, Median = 10, Range = 7
Sentence 1 โ Compare an average:
Class A has athan Class B, suggesting Class A.
Sentence 2 โ Compare the spread:
Class A has arangethan Class B, meaning Class A's results were.
โ๏ธ Exercise 1: Mean, Median, Mode & Range
Calculate all four measures. Round to 1 decimal place where needed.
โ๏ธ Exercise 2: Compare Two Datasets
Calculate the mean and range for each group, then answer the comparison questions.
โ๏ธ Exercise 3: Stem-and-Leaf Diagrams
Read the diagrams carefully and answer the questions.
๐ Exercise 4: Interpreting Frequency Charts
Read the comparative frequency chart and answer the questions.
โ๏ธ Exercise 5: Write Full Comparison Statements
Choose the correct words/values to build comparison statements. Every answer must include an average comparison AND a spread comparison.
๐ Practice Questions
Find the mean of: 3, 7, 9, 5, 11
Find the median of: 12, 4, 8, 15, 6, 10, 3
Find the mode of: 4, 7, 2, 7, 9, 3, 7, 1
Find the range of: 14, 3, 8, 19, 6, 11
A dataset is: 5, 8, 8, 9, 12, 15. Find the median.
Class A scores: 10, 12, 14, 16, 18. Class B scores: 5, 8, 13, 18, 21. Calculate the mean and range for each class.
Which class in Q6 had a higher mean? Which had a larger range? What does the range tell us about consistency?
Write a comparison statement for Q6 using both mean and range.
A back-to-back stem-and-leaf diagram shows: Team A leaves for stem 4: 2, 5, 7. Team B leaves for stem 4: 1, 3, 8. What are these values?
A dataset: 3, 5, 6, 6, 7, 8, 50. Identify the outlier and state its effect on the mean versus the median.
Dataset: 4, 6, 7, 7, 8, 9. Find the mean and median. Now add an outlier of 45. Recalculate both. How much did each change?
Describe a positively skewed distribution in terms of the tail and the relationship between mean, median and mode.
Two groups ran 100 m. Group X mean = 14.2 s, range = 3.1 s. Group Y mean = 13.8 s, range = 5.4 s. Which group was faster on average? Which was more consistent?
The heights (cm) of 7 plants: 12, 15, 18, 15, 22, 15, 19. Find all four measures.
Why is the median sometimes a better average than the mean?
Two shops record daily sales. Shop A: mean = ยฃ320, range = ยฃ80. Shop B: mean = ยฃ310, range = ยฃ200. Write a comparison statement.
A dataset has mean = 10 and median = 10. A value of 40 is added. Will the mean or median change more?
In a stem-and-leaf diagram, the stem is 3 and a leaf is 6. What value does this represent?
Describe what a symmetric distribution looks like and state the relationship between its mean, median and mode.
The range of Dataset A is 4 and the range of Dataset B is 14. What does this tell you about the two distributions?
Class A: mean = 70รท5 = 14, range = 18โ10 = 8. Class B: mean = 65รท5 = 13, range = 21โ5 = 16
Class A had a higher mean (14 vs 13). Class B had a larger range (16 vs 8), meaning Class B's scores were less consistent
"Class A has a higher mean (14) than Class B (13), suggesting Class A performed slightly better. Class A also has a smaller range (8) than Class B (16), showing Class A's results were more consistent."
Team A: 42, 45, 47. Team B: 41, 43, 48
Outlier = 50. Without it: mean โ 6.4, median = 6.5. With it: mean โ 13.4, median = 6.5. The mean is pulled up dramatically; the median barely changes.
Original mean = 41รท6 โ 6.8, median = (7+7)รท2 = 7. With 45: mean = 86รท7 โ 12.3 (changed by ~5.5), median = 7 (changed by 0)
Positively skewed: long tail to the right; Mean > Median > Mode
Group Y was faster on average (13.8 s < 14.2 s). Group X was more consistent (range 3.1 s < 5.4 s)
Ordered: 12, 15, 15, 15, 18, 19, 22. Mean = 116รท7 โ 16.6, Median = 15, Mode = 15, Range = 22โ12 = 10
The median is resistant to outliers โ it gives a more representative average when extreme values are present
"Shop A has a higher mean (ยฃ320) than Shop B (ยฃ310). Shop A also has a much smaller range (ยฃ80 vs ยฃ200), meaning Shop A's sales were far more consistent/predictable."
The mean will change much more. The median will shift only slightly (one position).
36
Bell-shaped, balanced about the centre. Mean = Median = Mode
Dataset A is much more consistent/clustered (spread of only 4); Dataset B is far more spread out/variable (spread of 14)
๐ Challenge: Multi-Step & Extended Problems
A teacher says: "The mean mark for my class was 62%." A student argues: "But the median was only 54%. Why is there such a big difference?" Explain, and state which measure is more representative. What does this suggest about the shape of the distribution?
Two basketball players' points per game over 8 games:
Player A: 12, 15, 18, 12, 20, 14, 11, 20
Player B: 8, 25, 6, 30, 7, 28, 5, 27
(a) Calculate the mean and range for each player. (b) Write a full comparison statement. (c) Which player would you prefer if your team needed consistent scoring? Justify using statistics.
The back-to-back stem-and-leaf diagram shows hours of TV watched per week by Year 7 and Year 8 students:
Year 7 | Stem | Year 8
9 5 2 | 1 | 3 6 8
8 6 4 1 | 2 | 0 2 5 9
5 3 | 3 | 1 4
(a) How many students are in each year group? (b) Find the median for each year group. (c) Write a comparison statement.
A dataset of 6 values has a mean of 12 and a range of 10. The smallest value is 7 and the largest is 17. One value is 8, another is 14. The mode is 15. Find all six values.
The ages of employees at two companies:
Company X: 22, 24, 28, 31, 34, 38, 42, 45
Company Y: 19, 20, 21, 22, 23, 45, 52, 60
(a) Calculate the mean and median age for both companies. (b) For Company Y, which is a better average โ mean or median? Justify by identifying the outliers. (c) Compare the ranges.
A frequency table shows maths test scores for Class 1 and Class 2 (scores out of 10):
Score: 4 5 6 7 8 9 10
Class 1 freq: 1 2 3 8 6 3 2
Class 2 freq: 4 3 5 5 3 3 2
(a) How many students in each class? (b) Find the modal score for each class. (c) Estimate the mean for each class. (d) Compare both classes fully.
A distribution is negatively skewed. Explain what this means in terms of: (a) the shape of the graph, (b) the relationship between mean, median and mode, (c) a real-life example where this might occur.
A sports coach claims: "Team A's mean score of 18 is better than Team B's mean score of 16." A statistician replies: "But Team A's result is misleading due to outliers." Team A data: 5, 6, 7, 8, 8, 9, 67. Team B data: 12, 14, 15, 17, 17, 18, 19. Verify both claims: calculate the mean for each team, identify any outliers, and determine which team's mean is more representative.
The mean is being pulled up by a few very high marks (outliers). The median, at 54%, is more representative as it is resistant to those extreme values. Since mean > median, the distribution is positively skewed โ most students scored below the mean, but a few high-scorers pulled it up.
(a) Player A: mean = 122รท8 = 15.25, range = 20โ11 = 9. Player B: mean = 136รท8 = 17, range = 30โ5 = 25. (b) "Player B has a higher mean (17 pts) than Player A (15.25 pts), but Player A has a much smaller range (9 vs 25), showing Player A is far more consistent." (c) For consistent scoring, choose Player A โ lower range means more predictable performance.
(a) Year 7: 3+4+2 = 9 students. Year 8: 3+4+2 = 9 students. (b) Year 7 ordered: 12, 15, 19, 21, 24, 26, 28, 33, 35 โ median = 24 hrs. Year 8 ordered: 13, 16, 18, 20, 22, 25, 29, 31, 34 โ median = 22 hrs. (c) "Year 7 has a higher median (24 hrs) than Year 8 (22 hrs), suggesting Year 7 students watched more TV on average."
Sum needed = 12 ร 6 = 72. Known values: 7, 8, 14, 17, 15, 15. Sum = 76. Adjust: replace one 15 with 11 โ values: 7, 8, 11, 14, 15, 17. Sum = 72 โ, range = 10 โ, mode = need reconsideration. With mode = 15: 7, 8, 14, 15, 15, 17 โ sum = 76 โ 72. Adjust lowest: 7, 8, 10, 15, 15, 17 โ sum = 72 โ. Values: 7, 8, 10, 15, 15, 17
(a) Co. X mean = 264รท8 = 33, median = (34+38)รท2? No โ ordered: 22,24,28,31,34,38,42,45 โ median = (31+34)รท2 = 32.5. Co. Y mean = 262รท8 = 32.75, median = (22+23)รท2... ordered: 19,20,21,22,23,45,52,60 โ median = (22+23)รท2 = 22.5. (b) Company Y has three high outliers (45, 52, 60) pulling the mean to 32.75, far above most workers' ages. The median (22.5) is more representative. (c) Co. X range = 45โ22 = 23; Co. Y range = 60โ19 = 41. Company Y has a much wider age spread.
(a) Class 1: 1+2+3+8+6+3+2 = 25 students. Class 2: 4+3+5+5+3+3+2 = 25 students. (b) Class 1 modal score = 7. Class 2 modal score = 6. (c) Class 1 mean = (4+10+18+56+48+27+20)รท25 = 183รท25 = 7.32. Class 2 mean = (16+15+30+35+24+27+20)รท25 = 167รท25 = 6.68. (d) Class 1 has a higher mean (7.32 vs 6.68) and higher mode (7 vs 6), suggesting Class 1 performed better overall.
(a) The graph has a long tail to the left; the bulk of data clusters to the right. (b) Mean < Median < Mode โ outliers on the left pull the mean down. (c) Example: exam scores where most students do very well but a few score very poorly (e.g. a hard re-sit paper).
Team A mean = (5+6+7+8+8+9+67)รท7 = 110รท7 โ 15.7. Team B mean = (12+14+15+17+17+18+19)รท7 = 112รท7 = 16. Team A has an outlier of 67 (far above the rest: 5โ9). Without it, Team A mean = 43รท6 โ 7.2. Team B's mean of 16 is more representative โ all values cluster between 12โ19 with no outliers. The statistician is correct: Team B's mean is more representative.