GCSE · Maths · OCR · Spec J560

Grouped data

Squash 200 different answers into five neat rows and the table looks lovely, but something has gone missing. What, and what can you still work out anyway?

Try it first

Build the row 40 ≤ x < 50

A grouped table has a row labelled 40 ≤ x < 50. Draw that row on the line: set where it starts and where it stops, then decide which ends get a filled circle (the end value counts) and which stay open (it doesn't). Ask yourself: is 40 in this row? Is 50?

Quick check

Which one is the modal class?

Books read last year by 33 pupils, grouped: 0 ≤ b < 10: 11 pupils 10 ≤ b < 20: 7 pupils 20 ≤ b < 30: 6 pupils 30 ≤ b < 40: 9 pupils

The highest frequency is 11, in the first row. Which of these is the modal class? Go with your gut, then say how sure you are.
How sure are you?

Finding the median class

Problem

48 pupils were asked how many texts they sent in one day, grouped: 0 ≤ n < 10: 7 10 ≤ n < 20: 15 20 ≤ n < 30: 18 30 ≤ n < 40: 8 Find the median class.

Your turn

Estimate the mean, one step at a time

30 pupils timed their journey to school (t minutes), grouped: 0 ≤ t < 10: 4 10 ≤ t < 20: 9 20 ≤ t < 30: 12 30 ≤ t < 40: 3 40 ≤ t < 60: 2 Estimate the mean journey time.

  1. We only know how many pupils are in each row, not their exact times. So we let the midpoint of each row stand in for every time inside it.
  2. missing step
Which line is step 2?

Two ways to estimate the range

Maximum possible rangevsAverage expected range

Same table, two different answers. Lowest interval 20 ≤ x < 30, highest interval 100 ≤ x < 110.

Focus

What you use

Maximum possible range

The upper bound of the highest interval and the lower bound of the lowest interval

Average expected range

The midpoint of the highest interval and the midpoint of the lowest interval

The insight

This is the bit people mix up: bounds go with the maximum possible range, midpoints go with the average expected range.

With this table

Maximum possible range

110 − 20 = 90

Average expected range

105 − 25 = 80

What it tells you

Maximum possible range

The biggest the range could possibly be, and it is never less than the true range

Average expected range

The range you would expect on average, with midpoints standing in for the missing values

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A grouped table is tidy, but it has quietly thrown the exact values away. Here's what you can still find out from it.

What you need to know

  • A grouped frequency table shows how many values fall in each interval (also called a group or class), not the values themselves.
  • The row 40 ≤ x < 50 holds values from 40 up to, but not including, 50. So 50 itself is excluded.
  • Have a goJas insists every number from 39 to 50 fits in the row 40 ≤ x < 50. Which of 39, 40, 49 and 50 prove Jas wrong?

    39 and 50

    39 is below 40, and 50 is excluded because of the < sign. The ≤ at the bottom keeps 40 in, and 49 is comfortably inside.

  • With whole-number data, that same row covers just 40, 41, and so on up to 49.
  • Once values are grouped, the exact value of each data point is lost, so the mean and range can only be estimated.
  • Add up the frequency column and you get the total number of data points.
  • Have a goA grouped table has frequencies 3, 8, 5 and 4. How many data points are in the data set?

    20

    The frequency column adds up to the number of data points. Counting the 4 rows would count rows, not data.

  • Intervals don't all have to be the same size, and neighbouring low-frequency intervals can be combined into one larger one.
  • The modal class is the interval with the highest frequency. Different groupings of the same data can give different modal classes.
  • Median class: work out the position (total frequency + 1) ÷ 2, then find where the running total first reaches it.
  • Have a goThe frequencies in a table add up to 40. What is the median position?

    20.5

    (40 + 1) ÷ 2 = 20.5. Halving 40 to get 20 forgets the + 1.

  • Estimate the mean: midpoint × frequency for each row, add them up, then divide by the total frequency.
  • Two range estimates: the maximum possible range uses the bounds, the average expected range uses the midpoints.

The big picture

A grouped frequency table gives the frequency of each interval instead of each value, so the exact values are lost. You can still read off the modal class and find the median class, and you can estimate the mean (midpoint × frequency, divided by the total frequency) and the range.

Key points

1A grouped table gives the frequency of each interval, so the exact values are no longer known.
240 ≤ x < 50 includes 40 and excludes 50.
3Modal class = the interval with the highest frequency, written as an interval.
4Median class = the interval whose running total first reaches (total frequency + 1) ÷ 2.
5Estimated mean = total of (midpoint × frequency) ÷ total frequency, never ÷ the number of rows.
6Range: maximum possible = upper bound of top interval − lower bound of bottom interval; average expected = midpoint of top − midpoint of bottom.

Worked example

Problem

20 pupils recorded their weekly reading time (t minutes), grouped: 0 ≤ t < 10: 3 10 ≤ t < 20: 9 20 ≤ t < 30: 6 30 ≤ t < 50: 2 Find the modal class, the median class and an estimate of the mean.

⚠ Watch out

Giving a frequency or a midpoint as the modal class (it should be an interval), and dividing the total of midpoint × frequency by the number of rows instead of by the total frequency.

🧠

Memory hook

Mean needs Midpoints: midpoint × frequency, add up, then divide by the total frequency, never by the number of rows.

✓

Check yourself

Without looking back: explain to an imaginary friend why 50 is not in the row 40 ≤ x < 50, and why a grouped table can never tell you the exact mean.

Flashcards

(11)
What does a grouped frequency table show?
The frequency of each interval (group or class) of values rather than of individual values, so a data set with a large variety of values fits without dozens of rows.
Which values does the row 40 ≤ x < 50 hold?
Values greater than or equal to 40 and less than 50, so 50 is excluded. For whole-number data that is 40 to 49.
What can't you know from a grouped frequency table?
The exact value of each data point. That is why the mean and range can only be estimated.
What does the sum of the frequency column tell you?
The total number of data points in the data set.
Must every interval in a table be the same size?
No. Sizes can vary, for example consecutive low-frequency intervals can be combined into one larger interval.
How do you give the modal class?
As the interval with the highest frequency, written in the same inequality form as the table, not as a frequency or a midpoint. Different groupings of the same data can give different modal classes.
How do you find the median class?
Find the running totals of the frequencies, work out the median position (total frequency + 1) ÷ 2 (for a total of 152 that is 76.5), then pick the interval whose running total first reaches it.
What are the steps to estimate a mean from a grouped table?
Midpoint of each interval (add the two bounds, divide by 2), multiply by its frequency, add these estimated totals, then divide by the total frequency.
An estimated total of £6,320 from 139 people: what is the estimated mean?
6,320 ÷ 139 ≈ £45.47. Dividing by the number of rows is wrong (6,320 ÷ 5 = 1,264).
How can you tell an estimated mean is believable?
It has to lie between the smallest and largest possible values, so an answer outside that span must be a mistake.
What are the two estimates of the range from a grouped table?
Maximum possible range = upper bound of the highest interval − lower bound of the lowest (never less than the true range). Average expected range = midpoint of the highest interval − midpoint of the lowest.

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.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More OCR GCSE Maths topics

How this lesson was checked. This OCR GCSE Maths (specification J560)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 9 October 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.