KS3 · Maths

Mean, median, and mode of a data set

Ask three people for the "average" star rating of a restaurant and you could get three different answers. All three can be right. Let's see how.

Have a go first

Where do the three averages sit?

Forty people gave a restaurant a rating from 1 to 5 stars. 1 star: 5 reviews. 2 stars: 9. 3 stars: 5. 4 stars: 9. 5 stars: 12. Drag a marker to where you think each average sits on the star scale, then check. No calculating yet. Trust your gut.

Predict, then check

Think about what stays the same while you move things around.

Two crates hold 18 and 12 packages. You move packages from the fuller crate to the other until both hold the same. How many packages do you move?

One method, three kinds of data

Problem

Find the mean from a list that includes a zero, from a frequency table, and from a frequency bar chart.

Maths · Reading data

Classroom temperature (°C): ten readings

Tap a bar to read it, then step through the tabs. Each bar is one reading, in the order they were taken. (The Range tab shows something else, so skip it today.)

061218243012345678910Reading number (in the order taken)Temperature (°C)

Raw data: one bar per reading. A line graph of these readings would plot the same ten values as ten points, one value each. Tap a bar to see its value.

Mean

20

Median

19.5

Mode

18

Range

9

Watch out: Two traps on a graph. The median is the middle of the readings once they are in order, not the middle bar and not halfway up the axis. The mode is the value that turns up most often, not the tallest bar.

Mark this answer

One line loses the mark

How many goals did a team score per match? 100 matches: 0 goals 19 times, 1 goal 34 times, 2 goals 21 times, 3 goals 15 times, 4 goals 11 times. Find the median. A pupil's answer is below. Lines 1 to 3 set the method up. Which line loses the mark?

A pupil's answer — which line goes wrong?

What do you think?

The mode on a bar chart

A frequency bar chart shows the star ratings from 40 reviews. The 5-star bar is the tallest and reaches 12. The 2-star bar and the 4-star bar are both 9 high.

Which is closest to what you think the mode is?
How sure are you?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Three ways to say "the middle" — and why you must say which one you mean.

What you need to know

  • An average is one single value that stands for the middle of a whole data set.
  • Mean, median and mode are three different averages, so always say which one you mean.
  • The mean is the total of all the values divided by how many values there are.
  • A zero adds nothing to the total, but it still counts as a value when you divide.
  • Have a goPriya scored 2, 0, 5 and 5 in four quizzes. She says: "The zero doesn't count, so my mean is 12 ÷ 3 = 4." Is she right?

    No. The mean is 12 ÷ 4 = 3.

    The zero adds nothing to the total, but it is still one of the four values, so it counts in the number you divide by.

  • From a frequency table, multiply each value by its frequency, add those up, then divide by the total frequency.
  • The median is the middle value once the data is in order, so order it first.
  • Have a goFind the median of 7, 2, 9, 4, 5. (Careful: the middle of the list as written is 9.)

    5

    In order the values are 2, 4, 5, 7, 9, and the median is the middle one of the ordered data, not the middle of the list as written.

  • With an even number of values, the median is the mean of the two middle ones.
  • The mode is the most common value, and it's the only average you can find for things like colours.
  • Have a goA class's favourite colours: red, blue, blue, green, red, blue. Which average can you find, and what is it?

    The mode: blue.

    Colours can't be totalled or put in order, so the mode is the only average, and blue is the most common.

  • In a table or bar chart, the mode is the value with the greatest frequency, not the frequency itself.
  • Every average lies within the data, but the mean doesn't have to be one of the values.

The big picture

Mean, median and mode are three different averages: each is one single value that stands for the middle of a whole data set. The mean evens the data out, the median is the middle of the ordered data, and the mode is the most common value. You can find them from a list, a line graph, a frequency table or a frequency bar chart, so long as you give the value the method points to and not a position, a frequency or a bar's height.

Key points

1Mean, median and mode are three different averages: each is one value for the middle of the data, so say which one you mean.
2Mean = overall total ÷ number of values. From a frequency table or bar chart, that number is the total frequency, not the number of rows or bars.
3Median = the middle of the ordered data (the mean of the two middle values when there is an even number). Its position is not its value.
4Mode = the most common value, not its frequency. Data can be bimodal, trimodal or have no mode.
5Every average lies within the data, but the mean, and the median of an even number of values, needn't be one of the data values.

Worked example

Problem

Seven pupils sent these numbers of texts in a day: 9, 3, 11, 3, 5, 8, 3. Find the mean, the median and the mode, and say which of the three is not one of the values.

⚠ Watch out

Grabbing a position, a frequency or a bar's height and calling it the average: dividing by the number of bars, saying "the median is the 10th value", or saying the mode is 12 when 12 is just the bar's height. Ask: which value does this point to?

🧠

Memory hook

MEAN = fair shares (even it out). MEDIAN = the middle once it's in order, like the median strip down the middle of a road. MODE = the one that turns up MOST.

✓

Check yourself

Cover the page. For 9, 3, 4, 3, 6, find the mode, median and mean. Which one isn't in the list? (Mode 3, median 4, mean 5: the mean.)

Flashcards

(15)
What is an average?
One single value that stands for the middle of a whole data set. Mean, median and mode are all averages, so say which one you mean.
12 blocks are shared evenly across 3 stacks, whatever the starting heights. What is the mean?
4 per stack. The total (12) and the number of stacks (3) don't change, so the level is 12 ÷ 3.
How do you work backwards from a mean to the total?
Total = mean × number of values. A mean of 6 conkers shared by 5 friends means 30 conkers in total.
Why must a zero be counted when you find a mean?
It adds nothing to the total but it is still a value, so it counts in the number you divide by. 9 years of frost data totalling 67 days gives 67 ÷ 9 ≈ 7.4 days, not 67 ÷ 8.
How do you find the mean from a frequency table?
Multiply each value by its frequency, add the subtotals for the overall total, then divide by the total frequency. Adding the value column is meaningless.
From a frequency bar chart, why divide by the sum of the bar heights and not the number of bars?
The bar heights are how many values there are. Dividing by the number of bars only gives the mean frequency per bar (40 ÷ 5 = 8), which isn't a mean rating.
A mean of 1.65 goals per match: is that possible?
Yes. The mean lies within the data but doesn't have to be one of the values. Sense check: it must be between the smallest and largest values, and it isn't rounded to a whole number.
What must you do before finding the median?
Put the data in numerical order, then take the middle value. Colours can't be ordered, so there is no median, and they can't be totalled, so there is no mean.
Two middle values are 42 and 45. What is the median? What if both are 29?
43.5, the mean of the two. If both middle values are 29, the median is 29. So the median isn't always a value in the data.
19 values, in order. Where is the median, and is that number its value?
In the 10th position, (19 + 1) ÷ 2. The 10th is a position, not the median's value. The median is the value you find there.
Where is the median on a line graph?
It is the middle of the ordered data values, not the middle of the axis. Readings of 18, 22, 23, 24 and 26 minutes in order have a median of 23.
What is the mode, and what is special about it?
The most common value. It's the only average you can find for non-numerical data like colours. Ordering a list first groups repeats, which makes it easier to spot.
What do bimodal and trimodal mean? Can there be no mode?
Bimodal: two modes. Trimodal: three. There's no mode when nothing appears more often than anything else.
On a frequency bar chart, what is the mode?
The value of the tallest bar, not its height. Equal bars that are shorter than the tallest don't affect the mode.
How do you find the mode on a line graph?
Look for the value with the most points at the same level. It's not the highest point. Three readings of 18 °C make 18 °C the modal temperature.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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