Mean, median, mode and range
Four ways to summarise a data set: the average, the middle, the most common, and the spread
About three to four lessons of 45 to 60 minutes
Which 'average' actually tells the truth?
A job ad says the 'average' salary at a company is $95,000. Sounds great, until you find out the boss earns $500,000 and the other nine staff earn $50,000 each. The mean (the number the ad used) is dragged upward by one huge value, even though almost nobody actually earns close to $95,000.
This is why statisticians never rely on just one 'average'. Mean, median and mode each answer a different question, and range tells you how spread out the data is. Once you can find all four and know which one to trust, you can see straight through a misleading headline.
- A company's 'average' salarythe mean can be pulled way up by one very high earner
- The most popular shoe size in a classthat is the mode, not the mean
- Race times to find the 'typical' runnerthe median ignores one very fast or very slow outlier
- A week of temperaturesthe range shows how much they varied, hot to cold
What students will be able to do
Students will calculate the mean, median, mode and range of a small numerical data set, explain what each measure tells you, and choose the most appropriate measure to describe a given set of data.
- I can calculate the mean of a data set by adding the values and dividing by how many there are.
- I can find the median by ordering the data and locating the middle value (or the average of the two middle values).
- I can find the mode as the value that appears most often, and know a set can have no mode, one mode, or several.
- I can find the range as the highest value minus the lowest value.
- I can explain which measure best describes a data set, especially when there is an outlier.
Standards this unit teaches
- AC9M7ST01Australian Curriculum v9 (ACARA)Mean, median, mode and range
Paraphrased: calculate the mean, median, mode and range for small sets of data, and use these measures to compare and interpret different numerical data sets.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Mean
- the total of all the values divided by how many values there are, often called 'the average'
- Median
- the middle value when the data is sorted from smallest to largest
- Mode
- the value that appears most often in the data set
- Range
- the difference between the highest and lowest values, showing how spread out the data is
- Outlier
- a value much higher or lower than the rest of the data, which can distort the mean
- Data set
- a collection of numbers, values or measurements collected for a purpose
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The mean: sharing the total equally
ConcreteImagine 5 friends have $2, $4, $4, $6 and $9 in coins. If they pooled all their money and split it perfectly evenly, how much would each person get? That equal-share amount is the mean.
To find the mean: add up every value to get the total, then divide by how many values there are. For the coins: 2 + 4 + 4 + 6 + 9 = 25, and there are 5 people, so 25 divided by 5 is 5. Each person would get $5 if it were shared equally, even though nobody actually had exactly $5.
- What two steps do you follow to find the mean?
- If 4 numbers add to 20, what is their mean?
2. The median: the true middle
ConcreteSort the same 5 values from smallest to largest: 2, 4, 4, 6, 9. The middle value, the one with exactly as many values above it as below it, is the median. Here it is 4, the third of five values.
With an odd number of values there is always a single middle one. With an even number of values, there is no single middle, so you average the two middle values. For 3, 5, 8, 10 (4 values), the two middle values are 5 and 8, and their mean, 6.5, is the median.
Find the median of 12, 3, 7, 3, 9.
- Sort the values from smallest to largest: 3, 3, 7, 9, 12.
- There are 5 values (odd), so the median is the middle (3rd) one.
- The 3rd value is 7.
Answer: The median is 7.
- Why must the data be sorted before you can find the median?
- With an even number of values, how do you find the median?
3. The mode and the range
PictorialThe mode is simply the value that shows up most often. In 2, 4, 4, 6, 9 the value 4 appears twice and everything else appears once, so the mode is 4. Some data sets have no mode (nothing repeats) and some have more than one (a tie for most frequent).
The range measures spread rather than a 'typical' value: it is the highest value minus the lowest. For 2, 4, 4, 6, 9 the range is 9 minus 2, which is 7. A small range means the data is tightly clustered; a large range means it is spread out.
Find the mode and range of 15, 22, 15, 30, 15, 22.
- Count how often each value appears: 15 appears 3 times, 22 appears 2 times, 30 appears once.
- The most frequent value, the mode, is 15.
- The range is the highest minus the lowest: 30 - 15 = 15.
Answer: Mode = 15, Range = 15.
- Can a data set have two modes? Give an example.
- What does a large range tell you about a data set?
4. Choosing the right measure
AbstractBack to the salary hook: mean, median and mode can each give a very different picture of the same data. When one value is far bigger or smaller than the rest (an outlier), the mean gets pulled toward it, but the median usually does not.
For the ten salaries $50k x 9 and $500k x 1: the mean is (9 x 50 + 500)/10 = $95k, dragged upward by the one outlier. The median (the middle value once sorted) is $50k, which better represents what a typical worker actually earns. This is exactly why real wage reports usually quote the median, not the mean.
A company has salaries (in thousands): 50, 50, 50, 50, 50, 50, 50, 50, 50, 500. Find the mean and the median.
- Mean: total is 9 x 50 + 500 = 950, divided by 10 people = 95.
- Median: sorted, the data is 50 (x9), 500. The two middle (5th and 6th) values are both 50, so the median is 50.
- The mean ($95k) is pulled far above what almost everyone earns; the median ($50k) reflects the typical salary much better.
Answer: Mean = $95k, median = $50k. The median is the fairer 'typical' figure here.
- Why does one very large value affect the mean more than the median?
- If a class of test scores has one student who scored 0 by mistake, which measure, mean or median, changes more?
Common misconceptions and how to address them
Misconception'Average' always means the mean.
Why it happens: Mean is the measure most often taught first, so students assume it is the only kind of average.
How to address it: Mean, median and mode are all types of 'average', each describing the data differently. Ask which one a claim is really using before trusting it.
MisconceptionTo find the median, you just look for the middle number without sorting first.
Why it happens: Students skip the sorting step and pick whichever value happens to sit in the middle of the list as given.
How to address it: The median is defined on SORTED data. Always order the values from smallest to largest first, then find the middle.
MisconceptionEvery data set has exactly one mode.
Why it happens: Students expect the mode to always exist and to always be unique, like the mean.
How to address it: A data set can have no mode at all (if nothing repeats), one mode, or several modes (a tie). Check by counting frequencies, do not assume.
MisconceptionThe range is the number of values in the data set.
Why it happens: Students confuse 'range' (a measure of spread) with 'how many values there are' (the count).
How to address it: The range is highest minus lowest, a single subtraction describing spread, not a count of items.
MisconceptionAn outlier should just be ignored when finding the mean.
Why it happens: Students learn the mean is sensitive to outliers and conclude the fix is to always throw the value away.
How to address it: An outlier changes the mean, but that does not make it wrong or safe to delete without reason. The better response is often to report the median alongside the mean, or investigate why the outlier occurred.
Guided practice (with answers)
1. Find the mean of 6, 10, 8, 12, 9.
Answer: 9, because 6+10+8+12+9 = 45, and 45 / 5 = 9.
2. Find the median of 18, 4, 9, 4, 11.
Answer: 9, because sorted the values are 4, 4, 9, 11, 18, and the middle (3rd) value is 9.
3. Find the mode of 3, 7, 3, 3, 5, 7.
Answer: 3, because it appears 3 times, more than any other value.
4. Find the range of 22, 5, 17, 30, 9.
Answer: 25, because 30 (highest) minus 5 (lowest) is 25.
5. Find the median of 4, 8, 6, 10 (an even number of values).
Answer: 7, because sorted the values are 4, 6, 8, 10, and the median is the mean of the two middle values, 6 and 8, which is 7.
6. A data set is 2, 2, 2, 2, 100. Which measure, mean or median, better describes a 'typical' value, and why?
Answer: The median (2), because the mean (21.6) is dragged far upward by the single outlier of 100.
Independent practice worksheets
Practise all four measures with computed, never-wrong answer keys, moving from Year 7 to more advanced Year 10 boxplot and comparison skills.
Differentiation
- Keep data sets small (5 values or fewer) and use whole numbers that divide evenly for the mean at first.
- Colour-code the sorted values before finding the median, so the 'middle' is visually obvious.
- Use a simple table with columns for mean, median, mode and range so students do not confuse which step they are on.
- Physically act it out: give students counters or coins so 'sharing equally' for the mean is tangible before it becomes a formula.
- Introduce data sets where the mean is not a whole number (e.g. requires one decimal place).
- Ask students to construct a data set with a given mean, median and mode all at once.
- Explore how adding one new value changes the mean, median, mode and range differently.
- Compare two real data sets (e.g. two classes' test scores) using all four measures to argue which class did better, and defend which measure is fairest to use.
Assessment: exit ticket
A three-question exit ticket sampling mean, median/mode, and interpreting an outlier.
1. Find the mean of 5, 9, 7, 11.
Answer: 8, because 5+9+7+11 = 32, and 32 / 4 = 8.
2. Find the median and mode of 3, 8, 3, 12, 3.
Answer: Median = 3 (sorted: 3, 3, 3, 8, 12, middle value is 3); Mode = 3 (appears 3 times).
3. A data set is 4, 5, 6, 5, 40. Which measure best describes a typical value, the mean or the median, and why?
Answer: The median, because the mean is pulled far upward by the outlier 40, while the median (5) reflects the other four close values.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 the mean (section 1), Lesson 2 the median (section 2), Lesson 3 mode and range (section 3), Lesson 4 choosing the right measure plus the exit ticket (section 4 and assessment).
- This unit assumes comfort with division (for the mean) and ordering numbers (for the median). Revisit the Grade 4 division and basic ordering skills if either is shaky.
- Language to keep repeating: mean is the equal share, median is the sorted middle, mode is the most frequent, range is the spread. Naming which one a question is asking for is half the battle.
- The number-line figure plots the raw values so students can see the mean sitting near the 'balance point' of the data, not just as an abstract division.
- Curriculum note: AC9M7ST01 (Australian Curriculum v9) introduces mean, median, mode and range at Year 7. This unit's independent practice also links to the Year 10 boxplot unit, since quartiles are a direct extension of the median.
- Present mode and print both work: use the Print button for a clean teacher copy or student handout, and project the page to teach straight from the number-line diagram.