Mean, Median, Mode & Spread
Match Measure to Definition
Draw a line to match each statistical measure to its definition.
Calculate the Mean
Find the mean of each data set.
2, 4, 6, 8, 10
15, 20, 25, 30, 10
3, 3, 5, 7, 7
Find the Median
Order the data set and find the median.
4, 1, 7, 3, 9
12, 8, 15, 10, 6, 14
2, 5, 5, 7, 8
Find the Mode
Find the mode of each data set.
3, 5, 3, 7, 3, 5
10, 12, 10, 14, 12, 10
A, B, A, C, B, A
Choose the Best Measure
Sort each situation into which measure of centre best represents the data.
Effect of Outliers
Which measure is MOST affected by an outlier?
Data set: 4, 5, 5, 6, 50. Adding 50 most changes the:
Which measure is LEAST affected by outliers?
For skewed data, the better measure of centre is usually the:
Real Data Analysis
Use the data set to answer all questions: Monthly rainfall (mm): 45, 60, 30, 90, 55, 40, 75, 60, 50, 80, 65, 70
Calculate the mean monthly rainfall.
Find the median.
Find the mode.
Find the range.
Which measure best represents this data? Explain.
Mean from a Frequency Table
Use mean = (sum of frequency × value) ÷ total frequency.
Value 2 appears 3 times, value 4 appears 5 times, value 6 appears 2 times. Mean = ?
Score 0: 2 students, Score 1: 4 students, Score 2: 4 students. Mean score = ?
Value 10 (freq 1), value 20 (freq 3), value 30 (freq 1). Mean = ?
Effect of Changing Data on the Mean
Draw a line to match each change to its effect on the mean.
Skewed Data — Mean or Median?
Sort each data description into whether mean or median is the better typical value.
Weighted Mean
A weighted mean accounts for different group sizes.
Class A (20 students) averaged 75. Class B (10 students) averaged 60. Overall mean?
3 tests weighted equally, scores 80, 90, 70. Mean = ?
Two assignments: first worth 40% scored 50, second worth 60% scored 80. Weighted mean?
Comparing Mean and Median
Answer these questions about choosing the right measure.
A data set is: 5, 6, 6, 7, 7, 8, 8, 9, 50. Calculate both the mean and median. Which is more representative? Why?
If you removed the value 50 from the data set above, how would the mean change? How would the median change?
Finding Missing Values Using the Mean
Work backwards from the mean to find a missing value.
Five values have a mean of 8. Four of the values are 6, 7, 9, and 10. What is the fifth value?
Three test scores have a mean of 75. Two of the scores are 68 and 80. What is the third score?
Missing Value Problems
Find the missing value given the mean.
4 values: 3, 7, 9, ?. Mean = 6. Missing value = ?
5 values with mean 10. Four values sum to 38. Fifth value = ?
3 values: 15, 20, ?. Mean = 18. Missing value = ?
Grouped Data — Estimated Mean
Estimate the mean from a grouped frequency table.
Interval 0–9: midpoint 4.5, freq 3; Interval 10–19: midpoint 14.5, freq 7; Interval 20–29: midpoint 24.5, freq 5. Estimate the mean using midpoint × frequency for each group.
Outliers and Their Effect on Measures
Investigate how outliers affect the three measures of centre.
Data set: 10, 12, 13, 14, 15, 16, 17, 100. Calculate the mean, median, and mode. Now remove 100 and recalculate. How much does each measure change?
Based on your investigation, which measure of centre is most resistant to the effect of outliers?
Mean, Median, Mode — Which to Use
Circle the most appropriate measure for each situation.
Reporting the 'typical' salary in Australia when a few executives earn very high salaries
Deciding which shoe size to stock most of in a sports shop
Reporting average rainfall when all values are similar
Finding the middle score in a ranked list of exam results
Measures of Centre in Sport
Apply measures of centre to a sports context.
A basketballer's points per game over 10 games: 15, 18, 12, 20, 8, 25, 14, 16, 19, 13. Find the mean, median, and range. Which measure best describes their 'typical' performance?
They score 45 in the next game (a career high). How does this affect the mean, median, and range? Which changes the most?
Mean Sequence
A data set grows by adding one more value each step. The value added is always 1 more than the previous. Find the missing mean.
Summarising Data — Full Analysis
Complete a full statistical summary for the given data set.
Data: number of texts sent per day by 15 students: 5, 12, 7, 45, 8, 9, 10, 6, 8, 11, 7, 8, 12, 3, 15. Find: (a) mean, (b) median, (c) mode, (d) range, (e) IQR. Which measure(s) do the outlier(s) affect most?
Skewness and Measures
Circle the correct relationship.
Data: 2, 3, 4, 5, 20 (right-skewed). Which is higher?
Data: 1, 10, 10, 11, 12 (left-skewed). Which is higher?
Perfectly symmetric data: which statement is true?
Statistical Measures in Context
Choose and justify the appropriate measure for each context.
A real estate agent wants to advertise the 'typical' house price in a suburb. The prices are: $450 000, $480 000, $490 000, $500 000, $510 000, $1 200 000. Which measure would be most favourable for advertising? Which is most honest? Show calculations.
Match Scenario to Best Measure
Draw a line from each scenario to the most appropriate measure of centre.
Transforming Data — Effect on Measures
Investigate how transforming data affects measures of centre and spread.
Data: 4, 6, 8, 10. Mean = 7, range = 6. If 3 is added to every value, what are the new mean and range?
If every value is multiplied by 2, what are the new mean and range?
In general: if every value is multiplied by k, the new mean = ___×original mean, and new range = ___×original range.
Sort Data Summaries
Sort each statement about a data set into 'Measures of Centre' or 'Measures of Spread'.
Weighted Mean in Real Life
Calculate and interpret weighted means.
School report: homework 20%, tests 50%, exam 30%. Scores: homework 85, tests 72, exam 68. Calculate the weighted mean score.
A student scores 90 on 3 tasks worth 1 point each and 50 on 1 task worth 3 points. Find the weighted mean (all worth their respective number of points).
Introduction to Standard Deviation
Calculate deviations from the mean.
Data: 4, 6, 8, 10, 12. Mean = 8. Calculate the deviation (value − mean) for each. Sum all deviations. What do you notice?
Now square each deviation and find the mean of the squared deviations. This is called the variance. Take the square root to get the standard deviation.
Real-World Investigation — Measures of Centre
Design and conduct a mini-investigation using measures of centre.
Collect data on a topic of your choice (e.g., how many steps you take each day for 10 days, or how long each TV show you watch lasts). Calculate mean, median, mode, and range. Which measure best describes your data? Are there any outliers?
Measures of Centre at Home
Investigate measures of centre using data from everyday life.
- 1Record the duration (in minutes) of 10 TV shows, movies, or YouTube videos you watch. Calculate mean, median, mode, and range.
- 2Weigh 10 items in your kitchen (tins, packets, etc.) in grams. Calculate the mean weight. Which item is the outlier?
- 3Count the number of pages in 10 different books on your bookshelf. Compare mean and median. Are they similar or different? What does this tell you?
- 4Challenge: Find the mean test score for 5 subjects over a term using your own results. Calculate a weighted mean if the subjects have different weightings.
Measures of Centre — Media Literacy
Evaluate how measures of centre are reported in the media.
A news article says 'The average Australian earns $90 000.' This uses the mean salary. Explain why the median salary (about $65 000 for Australia) would give a more realistic picture of most Australians' income.
Should news reports use mean or median when reporting house prices? Justify your answer with an example.
Interpreting Box Plots using Measures
A box plot shows: Min=5, Q1=15, Median=25, Q3=40, Max=70. Circle the correct answer.
IQR = Q3 − Q1 = ?
What percentage of data is between Q1 and Q3?
Is the distribution symmetric or skewed?
Measures and Fair Assessment
Apply measures of centre to a fair assessment scenario.
A student's test scores are: 45, 60, 72, 80, 88, 90, 35 (last one was when they were sick). Compare the mean with and without the sick score. What would be the fairest way to report this student's 'typical' performance?
Back-to-Back Comparison
Compare two data sets using measures of centre and spread.
Temperature (°C) in City A over 10 days: 18,20,19,21,22,18,20,19,21,20. City B: 10,15,22,28,25,13,18,30,12,27. Calculate mean and range for each city. Which city would you prefer to visit? Why?
Standard Deviation — Introduction
Explore standard deviation as a measure of spread.
Two classes take a test. Class A: 70,70,70,70,70 (all exactly 70). Class B: 60,65,70,75,80. Both have mean 70. Which class has a higher standard deviation? What does this tell us?
For Class B: calculate the deviation from the mean (value − 70) for each score. Square each deviation. Find the mean of the squared deviations (variance). Take its square root — this is the standard deviation.
Cumulative Mean Sequence
As values are added one by one, the mean changes. Fill in the missing cumulative mean.
Effect of Transformations on Measures
Circle the correct effect.
Every value in a data set is multiplied by 4. The IQR:
10 is subtracted from every value. The mean:
Every value is replaced by its double. The standard deviation:
Measures of Centre — Investigation
Design your own investigation using measures of centre.
Choose two groups to compare (e.g., Year 7 and Year 8 heights, or TV shows and movies). Plan how to collect data, which measures to calculate, and how to display and compare results. Include a hypothesis (prediction) before collecting data.
Measures of Centre — Extension Problem
Solve this challenging problem using measures of centre.
A student's mean score for 4 tests is 72. After the 5th test, the mean drops to 70. What did the student score on the 5th test?
In a class of 30 students, the mean height is 160 cm. 10 students leave and the remaining 20 have a mean height of 158 cm. What was the mean height of the 10 students who left?
Measures of Centre — Sort by Skewness
For each distribution, sort mean and median from largest to smallest.
Statistical Literacy — Conclusion
Reflect on what you have learned about measures of centre and spread.
In your own words, explain the key differences between mean, median, and mode. Give an example of when each would be the best measure to use.
Explain why the range alone is often insufficient as a measure of spread, and when the IQR is a better choice.
Calculating All Measures from Raw Data
Calculate all measures for the data set.
Data: 14, 9, 16, 12, 8, 15, 11, 14, 7, 12, 9, 14. Find: (a) mean, (b) median, (c) mode(s), (d) range. Show all working.
Median of Large Data Sets
Find the median position and value.
Data set has 15 values arranged in order. Median is at position:
Data set has 20 values arranged in order. Median is:
Ordered data: 3,5,7,9,11,13. Median = ?
Quartiles and Box Plots
Calculate quartiles and create a box plot.
Data: 5, 8, 9, 10, 12, 15, 17, 18, 20, 25. Find Q1, Q2 (median), Q3, IQR. Draw a box plot for this data.
Statistics Investigation — Sports Performance
Analyse a set of sports performance data.
A footballer's tackle counts across 14 games: 8,12,10,15,9,11,13,7,14,10,12,11,13,9. Find the mean, median, mode, and range. Draw a dot plot. Are there outliers? Is the footballer consistent?
Sort Measures by Resistance to Outliers
Sort each measure from most resistant to least resistant to the effect of outliers.
Normal Distribution — Circle Correct
Circle the correct answer about normal (bell-shaped) distributions.
In a normal distribution, approximately what % of data is within 1 standard deviation of the mean?
In a normal distribution, mean, median and mode are:
Which data set is MOST likely to follow a normal distribution?
Writing a Statistical Report
Write a brief statistical report for a data set.
Daily temperatures for two weeks: 22, 25, 19, 27, 24, 21, 23, 28, 20, 26, 23, 25, 22, 24°C. Write a 100-word statistical report including all measures of centre and spread. Comment on the consistency of temperature over the fortnight.
Running Mean Sequence
As values are added one at a time, track the running mean. Fill in the missing mean.
Reflection — What Have I Learned?
Reflect on your learning in this worksheet.
List the three measures of centre and one advantage and one disadvantage of each.
Describe a real-world situation where choosing the wrong measure of centre could be misleading. What should the correct measure be and why?