Printable · GCSE Foundation · ages 14-16
Statistics worksheet — GCSE Foundation
Fifteen questions across the statistics statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Statistics worksheet — GCSE Foundation
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- 1.A scatter graph has 12 plotted points. Four pupils each draw a line of best fit on the same graph and count how many of the 12 points lie above their line and how many lie below it: Amir — 10 above, 2 below. Priya — 6 above, 6 below. Kofi — 2 above, 10 below. Leah — 0 above, 12 below. Write down the name of the pupil whose line was drawn correctly, so that the points are roughly balanced above and below it.
- 2.On a scatter graph the horizontal axis shows height in centimetres and the vertical axis shows mass in kilograms. One point is plotted at (170, 65). Write down the height and the mass of that person.
- 3.A sector of a pie chart stands for 25% of the data. Work out the angle at the centre of that sector.
- 4.A scatter graph plots the number of years, x, that 20 employees have worked at a company against their salary, y. All the plotted points lie between x = 1 and x = 15. Write down the word used to describe an estimate for y made using a value of x that lies between 1 and 15.
- 5.A random sample of 50 pupils at a school were asked whether they prefer sport to music. 30 of the 50 pupils said they prefer sport. The school has 500 pupils altogether. Work out an estimate for the number of the 500 pupils who prefer sport.
- 6.A shop records the colour of every car that uses its car park one Tuesday. Its frequency table reads: black 44, silver 37, blue 26, red 18, white 12. Write down the frequency of black cars.
- 7.A quality inspector weighs a random sample of 50 packets of crisps from one day's production and finds their mean mass is 32.4 g. The factory makes 20,000 packets that day. Work out an estimate for the total mass, in kg, of all the packets made that day.
- 8.A council wants to find out what all residents of a town think about a new cycle lane. It posts a survey only on its website and asks people to fill it in online. Give a reason why this sample is likely to be biased.
- 9.Seven pupils were asked how many books they had read last month. Their answers were 3, 5, 5, 7, 8, 5, 3. Write down the mode.
- 10.A charity shop in Leicester made £2,400 in one month. A pie chart shows how this was spent: repairs took up an angle of 90°, wages took up an angle of 150°, and the rest was other costs. Work out how much was spent on wages.
- 11.Two classes sat the same maths test, both marked out of 100. Class A had a mean mark of 70 and a range of 30 marks. Class B had a mean mark of 70 and a range of 10 marks. Compare the marks of the two classes.
- 12.A teacher records the number of pets owned by each of 25 pupils in a class; each pupil owns 0, 1, 2, 3 or 4 pets. The teacher wants to show how many pupils own each number of pets. Write down the most suitable type of chart for this data, and give a reason for your answer.
- 13.A pie chart shows how 180 shoppers at a supermarket paid for their shopping. The sector for card payments has an angle of 90° at the centre. Work out how many of the 180 shoppers paid by card.
- 14.A timetable shows the departure times of a bus from Leeds to Harrogate: 08:15, 08:50, 09:25, 10:05, 10:40. Chloe arrives at the bus stop at 09:30. Work out how many minutes she has to wait for the next bus.
- 15.In a vertical line chart of test results, the tallest line stands for 40 pupils. The same results are to be shown in a pie chart of all 100 pupils. Work out the angle at the centre of the sector that will stand for those 40 pupils.
Answer key
- (a) Priya — A line of best fit should be drawn so that the plotted points are roughly balanced above and below it. Work out the difference between the two counts for each pupil: Amir 10 − 2 = 8; Kofi 10 − 2 = 8 (10 below and 2 above); Leah 12 − 0 = 12; Priya 6 − 6 = 0. Priya's line has the smallest difference, an exact balance of 6 above and 6 below, so her line is drawn correctly. Amir's line has 10 of the 12 points above it, so it is drawn too low. Kofi's line has 10 of the 12 points below it, so it is drawn too high. Leah's line has every single point below it, so it is not a line of best fit at all.
- (c) Height 170 cm, mass 65 kg — Method: a point on a scatter graph is written as a pair of coordinates in which the horizontal value is written first and the vertical value second, so each value is matched to the quantity named on its own axis. Working: in (170, 65) the value 170 is the horizontal coordinate and the horizontal axis shows height in centimetres, so the height is 170 cm; the value 65 is the vertical coordinate and the vertical axis shows mass in kilograms, so the mass is 65 kg. Answer: height 170 cm, mass 65 kg, each with the unit named on its own axis. The distractors: height 65 cm and mass 170 kg come from reading the pair the wrong way round, which would describe an impossible person; height 170 cm and mass 170 kg come from reading the horizontal coordinate for both quantities and never using the second number; height 235 cm and mass 105 kg come from combining the two coordinates, 170 + 65 and 170 − 65, instead of reading them separately.
- (c) 90° — Method: the sectors of a pie chart share the 360° at the centre of the chart in the same proportion as the data, so a sector's angle is its share of the data multiplied by 360°. Working: a share of 25% is the fraction 25/100, which is one quarter of the whole chart, and one quarter of 360° is 360 ÷ 4 = 90°. Answer: 90°, an angle in degrees rather than a percentage. The distractors: 25° comes from sharing out 100 instead of 360, so the percentage is written straight down as a number of degrees; 14.4° comes from dividing 360 by 25 instead of multiplying 360 by the fraction 25/100, which is the division done the wrong way round; 45° comes from taking a quarter of 180° instead of a quarter of 360°, treating the pie chart as a semicircle.
- (b) Interpolation — The salary is being estimated for a value of x between 1 and 15, which is inside the range of x-values that were actually plotted, so this is interpolation. Extrapolation would apply if the estimate used a value of x below 1 or above 15, outside the plotted range. Correlation describes the relationship between the two variables, not the reliability of an estimate, and causation describes one variable actually causing a change in the other, which is a different idea altogether — neither is the word being asked for here.
- (d) 300 pupils — Method: an estimate for a whole population is made by finding the proportion in the sample and applying that same proportion to the population. Working: in the sample 30 of the 50 pupils prefer sport, a proportion of 30 ÷ 50 = 0.6, and applying that proportion to the school gives 0.6 × 500 = 300 pupils. Answer: 300 pupils, and it is only an estimate, because a different random sample of 50 would give a slightly different figure. The distractors: 200 pupils comes from scaling up the 20 pupils in the sample who did not prefer sport, 20 × 10, which answers the opposite question; 150 pupils comes from reading 30 out of 50 as 30% and taking 30% of 500; 60 pupils comes from working out the proportion correctly as 60% and then writing the 60 down as a number of pupils instead of applying it to the 500.
- (d) 44 — Method: the frequency of a category in a frequency table is the number of times that category was counted, and it is read from the row for that category. Working: the rows of the table pair each colour with its count, and the row for black is paired with the count 44, so the frequency of black cars is 44. Answer: 44 cars — a frequency is a count of cars, not a colour and not a percentage. The distractors: 37 comes from reading the count paired with silver, that is from reading the wrong row of the table; 137 comes from adding every count in the table, 44 + 37 + 26 + 18 + 12, which gives the total number of cars rather than the frequency of one colour; 5 comes from counting how many different colours the table lists instead of how many cars were black.
- (d) 648 kg — Method: to estimate a total from a sample, multiply the sample's mean by the number of items in the whole population, then check the units the question asks for. Working: 32.4 g × 20,000 = 648,000 g. Converting to kilograms, 648,000 ÷ 1,000 = 648 kg. This is only an estimate, not an exact total, because it assumes every one of the 20,000 packets has exactly the sample mean mass, when in reality individual packets vary above and below it. Giving 1.62 kg multiplies the mean by 50, the SAMPLE size, instead of by 20,000, the number of packets actually made that day — this finds the total mass of the 50 sampled packets, not the day's production. Giving 32.4 kg treats the sample mean itself, in grams, as if it already were the day's total mass in kilograms, skipping the scaling up altogether. Giving 648,000 kg correctly scales the mean up to the whole day's production but never converts the answer from grams to kilograms, leaving it 1,000 times too large. Always scale a sample's mean up by the SIZE OF THE WHOLE POPULATION, and always finish by checking the units the question asks for.
- (d) Only internet users reach the website; others are excluded. — Method: a sample is biased when it systematically leaves out part of the population, or systematically over-represents another part. Working: anyone without internet access, or who does not visit the council's website, has NO chance of being included — the sample is drawn only from internet-using residents, which is not the whole town. Saying too many people might respond because the survey is free confuses bias with sample size — bias is about who CAN be reached, not how many respond. Saying people might lie describes a different problem, response honesty, not who was sampled in the first place. Saying online surveys cannot be anonymous is not a reason connected to bias at all. A sample is biased when part of the population has no chance of being included, whatever the reason for that.
- (c) 5 — Method: the mode is the value that occurs most often, so count how many times each different value appears and compare the counts. Working: 3 appears twice, 5 appears three times, 7 appears once and 8 appears once, so the highest frequency is three and the value carrying it is 5. Answer: 5. The distractors: 3 comes from writing down the frequency of the most common answer instead of the answer itself; 7 comes from taking the middle number of the list as it was written, which applies the median without ordering the data and without answering the question asked; 8 comes from picking the largest value, which confuses the mode with the maximum.
- (b) £1,000 — Wages take up 150° out of 360°, so the amount spent on wages is 150 ÷ 360 × 2400 = £1,000. Choosing £600 uses the repairs angle, 90°, instead of the wages angle: 90 ÷ 360 × 2400 = 600. Choosing £3,600 treats the angle in degrees as if it were a percentage, 150 ÷ 100 × 2400 = 3600, instead of dividing by 360°. Choosing £800 uses the angle for the 'other costs' sector, 360 − 90 − 150 = 120°, instead of the wages sector: 120 ÷ 360 × 2400 = 800.
- (a) The means are equal, and Class B's marks are the more consistent because its range is smaller. — Method: comparing two distributions needs two things — a measure of average and a measure of spread — and each must be put into the context of the question. Working: both classes have a mean mark of 70, so on average the two classes scored the same; the range measures spread, and Class A's range of 30 marks is three times Class B's range of 10 marks, so Class B's marks sit closely around the mean while Class A's are far more spread out. Answer: the means are equal, and Class B's marks are the more consistent because its range is smaller. The distractors: the reply crediting Class A with more consistency reverses the meaning of the range, treating a larger range as tighter data when a larger range means more spread; the reply that Class A's mean mark is higher compares the wrong pair of figures, reading the range of 30 as an average; the reply that Class B's mean mark is higher reads the spread correctly but its claim about the means is false, since both means are 70.
- (a) A vertical line chart (discrete numerical data) — The number of pets is discrete numerical data — whole-number values such as 0, 1, 2, 3 or 4 — recorded for one variable, so a vertical line chart is the chart specified for this kind of data. A bar chart is used for categorical data, such as favourite colour, not numerical values counted like this. A pie chart shows proportions of a whole and does not show the frequency of each separate value. A scatter graph compares two different variables against each other, and only one variable, the number of pets, is recorded here.
- (d) 45 — Method: convert the angle into a fraction of the full circle, 360°, then apply that fraction to the total number of shoppers. Working: the card sector is 90° out of 360°, a fraction of 90 ÷ 360 = 0.25. Applying that fraction to the 180 shoppers gives 0.25 × 180 = 45 shoppers. Giving 90 states the angle itself, not a number of shoppers — the angle first has to be converted into a fraction. Using the remaining angle, 360 − 90 = 270°, and scaling that, 270 ÷ 360 × 180 = 135, finds the number who did NOT pay by card, not the number who did. Dividing 360 by 90, 360 ÷ 90 = 4, finds how many equal 90° sectors fit in the circle, a fact about the pie chart's shape, not about the shoppers at all. Always convert the angle to a fraction of 360° first, and apply that same fraction to the total number of people.
- (a) 35 minutes — Chloe arrives at 09:30, so she needs the first departure time that is at or after 09:30. The bus at 09:25 has already left five minutes before she arrives, so it is missed, not caught; the next one on the timetable is 10:05. From 09:30 to 10:00 is 30 minutes, then a further 5 minutes to 10:05, so the wait is 35 minutes. Giving 40 minutes measures the gap from the missed 09:25 bus to the 10:05 bus, counting from the bus Chloe missed instead of from the time she arrived. Giving 5 minutes uses the missed 09:25 bus, working out 09:30 − 09:25 the wrong way round. Giving 45 minutes comes from treating the times as decimals — 10.05 − 9.30 = 0.75, then × 60 = 45 — instead of working in hours and minutes, where 05 minutes past the hour is not 0.05 of an hour.
- (a) 144° — Method: a pie chart shares the 360° at its centre between the categories in proportion to their frequencies, so the angle of a sector is that category's fraction of the total multiplied by 360°. Working: the sector stands for 40 pupils out of 100, which is the fraction 40/100; one pupil is worth 360 ÷ 100 = 3.6°, so 40 pupils are worth 40 × 3.6° = 144°. Answer: 144°, and the answer is an angle in degrees, not a number of pupils. The distractors: 40° comes from sharing out 100 instead of 360, so the percentage is written straight down as a number of degrees; 216° comes from working out the angle for the other 60 pupils, 60 × 3.6°, which is the rest of the pie chart; 180° comes from assuming that the tallest line must stand for half of the pupils and so take half of the chart.
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