Cheat sheet: Statistics
Reading charts
- Read the title, labels and scale first.
- Check what each step on the scale is worth.
- Pictogram: each symbol has a value; half a symbol is half the value.
Tables and timetables
- Rows go across, columns go down.
- Two-way table: rows and columns add to the totals.
- Journey time = arrive − leave (count up to the next hour).
Line graphs
Good for change over time. Steeper = faster change. Flat = no change. Read across to the axis for the value.
Pie charts
- The whole circle is 360°.
- Degrees per item = 360 ÷ total.
- Sector size shows a share, not a number.
Mean
Mean = total ÷ how many. Mean of 2, 4, 6, 8: 20 ÷ 4 = 5.
Reverse: total = mean × how many. Then subtract the known numbers.
Mode, median, range
- Mode: the most common value.
- Median: the middle value after ordering (two middles: halfway).
- Range: biggest − smallest.
Venn and Carroll
Venn: overlap = both. Outside the circles = neither. Carroll: a grid with yes/no for each property.
Total = only A + both + only B + neither.
Words to know
data, frequency, axis, scale, key, pictogram, mean, average, mode, median, range, sector, timetable, interval.
Mini lessons
Pictograms and bar charts
Skills: STAT-01
A pictogram uses pictures, and a key tells you what each picture is worth. A bar chart uses the height of bars. Always check the scale: the lines may go up in 2s, 5s or 10s, not 1s.
In a pictogram, each ★ stands for 4 books. Ruby’s row has 3 whole stars and half a star. How many books is that? In the bar chart above, how many more children chose dog than fish?
- 3 whole stars: 3 × 4 = 12 books.
- Half a star is half of 4 = 2 books.
- 12 + 2 = 14 books.
- Bar chart: dog is 8 and fish is 3. 8 − 3 = 5.
Answer: 14 books, and 5 more children
Tables and two-way tables
Skills: STAT-02
A table organises data in rows and columns. A two-way table sorts people by two things at once. The totals let you find missing numbers by adding or subtracting.
Rows and columns each add up. The missing number is in a row with 7 and a total of 20.
40 children were asked how they get to school. Use the table to find how many boys come by bus.
- Boys in total: 20. Boys who walk: 7.
- The rest come by bus: 20 − 7 = 13.
- Check with the column: girls by bus 9 + boys by bus 13 = 22, the bus total ✓
Answer: 13 boys
Line graphs
Skills: STAT-03
A line graph shows how something changes, such as height or temperature over time. The points are joined by lines. A steeper line means a faster change.
The line is steepest between week 1 and week 2, where the plant grew from 5 cm to 9 cm.
Use the graph. In which week did the plant grow the most? How tall was it after 3 weeks?
- Heights at weeks 0 to 4 are 2, 5, 9, 12, 14 cm.
- Growth each week: 3, 4, 3, 2 cm.
- The biggest growth is 4 cm, in week 2 (from week 1 to week 2).
- After 3 weeks, the graph shows 12 cm.
Answer: Week 2 (4 cm); 12 cm
Timetables and data tables
Skills: STAT-04
In a timetable, each column is a journey and each row is a stop. To find how long a journey takes, subtract the leaving time from the arriving time. Counting up to the next hour is an easy way.
Read down a column to follow one bus. Read across a row to see every bus at one stop.
How long does Bus A take to go from Oak Road to School? Which bus must Mo catch from Oak Road to arrive at School by 09:00?
- Bus A leaves Oak Road at 08:15 and reaches School at 08:52.
- From 08:15 to 08:52: 08:15 → 08:30 is 15 min, then to 08:52 is 22 min. 15 + 22 = 37 minutes.
- Bus A arrives 08:52 (before 09:00). Bus B arrives 09:22 (too late).
Answer: 37 minutes; Bus A
Pie charts
Skills: STAT-05
A pie chart shows how a whole is shared. The whole circle is 360°. Work out how many degrees one item gets by dividing 360 by the total number of items.
40 children share 360°, so each child is 360 ÷ 40 = 9°. Banana: 20 × 9 = 180°.
40 children chose a favourite fruit. How many degrees is the banana sector (20 children)? A different pie chart of 30 children has a sector of 72°. How many children is that?
- 360 ÷ 40 = 9° for each child.
- Banana: 20 × 9 = 180°.
- For 30 children, the sector is 72° out of 360°: 72 ÷ 360 = 1/5.
- 1/5 of 30 = 6 children.
Answer: 180°, and 6 children
Mean and reverse mean
Skills: STAT-06
The mean is the “fair share” value: add all the numbers and divide by how many there are. To go backwards, multiply the mean by how many numbers to get the total, then subtract the numbers you know.
The mean of five numbers is 8. Four of them are 6, 7, 9 and 10. What is the fifth number?
- Total of five numbers = 8 × 5 = 40.
- The four known numbers: 6 + 7 + 9 + 10 = 32.
- The fifth: 40 − 32 = 8.
- Check: (32 + 8) ÷ 5 = 8 ✓
Answer: 8
Mode, median and range
Skills: STAT-07. Stretch: usually taught in Year 7, but some 11+ tests include it.
The mode is the value that appears most often. The median is the middle value when the numbers are in order. The range is the biggest number minus the smallest, and it tells you how spread out the data is.
Always put the numbers in order first.
Find the mode, median and range of 5, 8, 3, 9, 5, 12, 6, 5.
- Put in order: 3, 5, 5, 5, 6, 8, 9, 12.
- Mode: 5 appears three times.
- There are 8 numbers, so the median is halfway between the 4th and 5th: 5 and 6. Halfway is 5.5.
- Range: 12 − 3 = 9.
Answer: Mode 5, median 5.5, range 9
Venn and Carroll diagrams
Skills: STAT-08
A Venn diagram sorts things into overlapping circles. The overlap is for things in both groups, and the outside is for things in neither. A Carroll diagram is a grid with yes and no boxes.
The middle is for children in both groups. The outside is for children in neither group.
30 children: 14 like football, 11 like swimming, and 5 like both. How many like neither?
- Football only: 14 − 5 = 9.
- Swimming only: 11 − 5 = 6.
- In a circle: 9 + 5 + 6 = 20.
- Neither: 30 − 20 = 10.
Answer: 10 children
Worked examples across the levels
Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.
Level 1 (Year 3): one step
In a pictogram, each smiley stands for 2 children. There are 5 smileys. How many children is that?
- 5 × 2 = 10.
Answer: 10 children
Level 2 (Year 4): one or two steps
Books read: Monday 6, Tuesday 9, Wednesday 4. How many books were read in the three days?
- 6 + 9 = 15.
- 15 + 4 = 19.
Answer: 19 books
Level 3 (Year 5): two steps
Find the mean of 4, 7, 9 and 12.
- Total: 4 + 7 + 9 + 12 = 32.
- Divide by 4: 32 ÷ 4 = 8.
Answer: 8
Level 4 (exam standard): several steps or working backwards
The mean of five test scores is 14. Four of the scores are 12, 15, 13 and 18. What is the fifth score?
- Total of five: 14 × 5 = 70.
- Four known: 12 + 15 + 13 + 18 = 58.
- Fifth: 70 − 58 = 12.
Answer: 12
Level 5 (challenge): three to five steps
The mean of three numbers is 15. A fourth number is added and the mean becomes 17. What is the fourth number?
- First total: 15 × 3 = 45.
- New total: 17 × 4 = 68.
- Fourth number: 68 − 45 = 23.
- Check: (45 + 23) ÷ 4 = 17 ✓
Answer: 23
Common mistakes
Giving the total as the mean
Mistake code: DATA_MEAN_AS_TOTAL
The mistake: Saying the mean of 2, 4 and 6 is 12 (the total).
Why it is wrong: The mean is a fair share. You must divide the total by how many numbers there are.
How to avoid it: Check your answer: the mean must sit between the smallest and the largest number.
Reading every gridline as 1
Mistake code: MEAS_SCALE_STEPS_AS_ONE
The mistake: A bar halfway between the 2 and 4 lines is read as 2.5, or a bar on the 6 line is read as 3.
Why it is wrong: On a scale that goes up in 2s, each line is worth 2, and halfway is 1 more.
How to avoid it: Work out what each gridline is worth before you read a bar.
Finding the median without ordering
The mistake: Taking the middle of 7, 2, 9, 4, 5 as 9 because it is in the middle of the list.
Why it is wrong: The median is the middle value after the numbers are put in order.
How to avoid it: Write the numbers in order, smallest first, then cross off from both ends.
Reading a pie chart sector as a number
The mistake: Saying that a 90° sector in a chart of 32 children means 90 children.
Why it is wrong: Degrees show a share of 360°, not a count.
How to avoid it: Turn the sector into a fraction of 360°, then find that fraction of the total.
Quick check
Five questions, easy to hard. Use paper and have a go before you open the answer.
In this pictogram, each star stands for 4 apples. Orla’s row has 6 whole stars and half a star (dotted). How many apples is that?
Show answer
26 apples
6 × 4 = 24. Half a star is 2. 24 + 2 = 26.
Find the median of 8, 3, 11, 6 and 4.
Show answer
6
In order: 3, 4, 6, 8, 11. The middle one is 6.
36 children were asked about their favourite sport. The football sector of the pie chart is 120°. How many chose football?
Show answer
12 children
120° out of 360° is 1/3. 1/3 of 36 = 12.
A train leaves at 10:48 and arrives at 12:09. How long is the journey?
Show answer
1 hour 21 minutes
10:48 to 11:00 is 12 min. 11:00 to 12:00 is 60 min. 12:00 to 12:09 is 9 min. 12 + 60 + 9 = 81 min = 1 h 21 min.
The mean of five numbers is 11. Four of them are 9, 14, 10 and 13. What is the fifth number?
Show answer
9
Total = 11 × 5 = 55. Known: 9 + 14 + 10 + 13 = 46. Fifth: 55 − 46 = 9.
For grown-ups
- Make a quick survey at home, such as favourite pizza toppings, and draw a bar chart together. Ask “what does the scale go up in?”
- Look at charts in newspapers or on the news and ask your child to explain what they show in one sentence. Ask what the title and labels say first.
- For the mean, use sweets or coins: share them out equally between cups. This is the “fair share” idea behind the mean.