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.
Answer key: Statistics worksheet — GCSE Foundation
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- (d) £15.50 — 2 × 6 = 12, and 12 + 3.50 = 15.50, so the estimated fare is £15.50. Choosing £12.00 stops after 2 × 6 = 12 and forgets to add the £3.50. Choosing £19.00 adds the distance and the constant first and then multiplies: 6 + 3.50 = 9.50, and 9.50 × 2 = 19.00, applying the ×2 to the whole sum instead of only to the distance. Choosing £13.00 multiplies only the constant term by 2 instead of the distance: 2 × 3.50 = 7, and 7 + 6 = 13.00.
- (b) A person's shoe size and their favourite colour — A person's shoe size is not linked to which colour they prefer, so these two show no correlation. The other three pairs are all genuinely correlated: distance travelled and fuel used rise together, which is positive correlation; hours of revision and test score generally rise together, which is also positive correlation; and as outdoor temperature rises, fewer woolly hats are sold, which is negative correlation. Negative correlation is still a real relationship between two variables — it is not the same thing as no relationship at all, so the temperature and hats pair is not the answer to this question.
- (d) 19 kg — Method: estimate the mean of grouped data by multiplying each class's midpoint by its frequency, adding the four totals, then dividing by the total frequency. Working: the midpoints are 5, 15, 25 and 35 kg. The weighted totals are 11 × 5 = 55, 5 × 15 = 75, 5 × 25 = 125 and 9 × 35 = 315, which add to 570. Dividing by the 30 dogs gives an estimate of 570 ÷ 30 = 19 kg. Giving 5 kg reads off the midpoint of the modal class, 0 < m ≤ 10, the class with the most dogs — but the class with the most dogs is not where the mean falls, and neither is a substitute for actually calculating it. Giving 20 kg averages the four midpoints, (5 + 15 + 25 + 35) ÷ 4, treating every class as equally likely and ignoring that far more dogs are in the lightest and heaviest classes than in the middle two. Giving 570 kg stops after finding the correct weighted total and forgets the final division by the 30 dogs. Always weight each midpoint by its own frequency, and always finish by dividing by the total frequency, not the number of classes.
- (d) 30 — Method: on a dual bar chart each bar is a separate frequency, so a total for the whole class is found by combining the two frequencies the bars show. Working: the bars show 14 boys and 16 girls, and 14 + 16 = 30. Answer: 30 pupils, a count of pupils in the class. The distractors: 32 comes from doubling the taller bar, 16 + 16, as though the two bars were equal; 28 comes from doubling the shorter bar, 14 + 14, in the same way; 2 comes from finding the difference between the two bars, 16 − 14, which answers how many more girls there are rather than how many pupils there are.
- (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) 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.
- (a) 62.5 — Method: a mean cannot be averaged with a new value — rebuild the total, add the new value to it, then divide by the new count. Working: three numbers with a mean of 50 have a total of 50 × 3 = 150; adding 100 makes the total 150 + 100 = 250; there are now 4 numbers, so the new mean is 250 ÷ 4 = 62.5. Answer: 62.5. The distractors: 75 comes from averaging the old mean with the new value, (50 + 100) ÷ 2, which ignores that three numbers pull against one; 50 comes from assuming an extra value leaves the mean unchanged; 37.5 comes from dividing the old total of 150 by the new count of 4, adding the new value to the count but not to the total.
- (d) Only families with strong feelings bothered to reply. — Method: a survey has non-response bias when only some of the people asked actually reply, and those who do are not a typical cross-section of everyone who was asked. Working: only 30 of the 200 families sent back their questionnaire, and 27 of those 30 — the great majority — said they were unhappy. Families who feel strongly about an issue, particularly those with a complaint, are far more likely to make the effort to reply than families who are simply satisfied and see no need to say anything, so the 30 replies over-represent unhappy families. Saying the families who replied were picked at random by the school gets the sampling the wrong way round: nobody picked them — they picked themselves by deciding to reply, and that is precisely why they are not a typical cross-section of all 200. Saying postal surveys always have low response rates restates that the response was low without explaining why a low response rate, on its own, makes a result unrepresentative — it is the reason FOR the low response, not the low response itself, that causes the bias here. Saying that the 27 unhappy replies show most families are unhappy is exactly the mistake the question is warning against: it treats the loudest 30 replies as if they stood for the other 170 who never sent theirs back. A low response rate is a warning sign only because the people who bother to reply are rarely typical of everyone who was asked.
- (c) Thursday — Method: on a bar chart the tallest bar belongs to the greatest value, so compare the five temperatures and then read off the day that the greatest one belongs to. Working: the temperatures are 20 °C, 22 °C, 18 °C, 25 °C and 23 °C; in order of size these are 18, 20, 22, 23 and 25, so the greatest temperature is 25 °C, and the day recorded with 25 °C is Thursday. Answer: Thursday — the answer is a day, not a temperature. The distractors: Friday comes from stopping at the final temperature listed instead of comparing all five; Wednesday comes from picking out the shortest bar, 18 °C, and so answering for the lowest temperature rather than the highest; Monday comes from writing down the first day in the chart without comparing any of the temperatures at all.
- (c) No, the mode here is the lowest value of the nine — Method: an average is meant to stand for the data as a whole, so test any proposed average by asking how many values it sits near. Working: the value 4 appears three times and every other count appears once, so 4 is indeed the mode. But those three hours are the quiet ones at the start of the day, and the other six counts run from 11 up to 25; putting the nine counts in order, the middle one is the fifth, which is 13. So the mode sits at the very bottom of the data, with six of the nine hours far above it. Answer: no, because the mode here is the lowest value of the nine, so it describes the quiet opening hours rather than a typical hour. The distractors: saying the mode can only be used when no value repeats reverses the definition, since a mode exists only because a value does repeat; saying the mode is the value that occurs most often is a correct definition, but being the commonest value does not make a value typical when it lies at one end of the data; saying the mode is the best average for any list of numbers ignores the fact that mean, median and mode each describe a population well in different circumstances.
- (d) Full-time workers, as most are at work at that time — Method: a sample is biased when the method of contact makes part of the population much less likely to be reached, so test each group against where its members actually are between 10 am and 2 pm on a weekday, and test each stated reason against the facts. Working: those hours are the middle of the working day, so people in full-time employment are at work and not beside a landline telephone, while people who are retired and people who are unemployed are far more likely to be at home and are reached at the usual rate; the method therefore collects far fewer replies from full-time workers than their share of the adult population the council is consulting. Answer: full-time workers, as most are at work at that time. The distractors: the reply naming retired people rests on the false claim that most retired people are at work in the daytime, when in fact a daytime call reaches them more easily than anyone; the reply naming unemployed people rests on the false claim that they are out during the day, when they too are among the easiest people to reach by a daytime call; the reply naming children rests on the false claim that children are at home at 11 am on a Tuesday, when they are at school and so are not reached by the call at all, and school-age children are in any case not the adults whose views the council is collecting.
- (b) Tea — Method: the mode is the category with the highest frequency. Working: tea has a frequency of 8, which is higher than coffee's 7, hot chocolate's 4 and juice's 1. Mode = tea. Coffee has the second-highest frequency, 7, not the highest, so it is not the mode. Hot chocolate, with a frequency of 4, and juice, with a frequency of 1, both have lower frequencies still. Always compare every frequency in the table before naming the mode — the highest number wins, however close the next one is.
- (c) Strong negative correlation — Method: correlation is described by two things — the direction the points take as the graph is read from left to right, and how closely the points lie to a single straight line. Working: reading the pairs in order of age, the ages rise 14, 18, 23, 27, 31, 36, 42, 49 while the scores fall 92, 88, 85, 80, 78, 74, 70, 65; the score falls at every single step, with no reversal anywhere, so the points fall from left to right and lie close to a straight line. Answer: strong negative correlation — negative for the falling direction, strong because every point follows the pattern. The distractors: strong positive correlation comes from noticing a clear pattern and calling any clear pattern positive, without checking the direction; weak negative correlation comes from reading the direction correctly but judging points that do not lie exactly on a straight line to be only loosely related, when these eight fall without a single exception; no correlation comes from reading a falling trend as though it showed no relationship at all, when a falling trend is itself a relationship.
- (d) 20 — 0.4x + 1 = 9, so 0.4x = 9 − 1 = 8, and 8 ÷ 0.4 = 20, so 20 sessions are needed. Choosing 23 divides 9 by 0.4 without first subtracting the 1: 9 ÷ 0.4 = 22.5, rounded up to 23. Choosing 25 subtracts the wrong way, adding the 1 instead of taking it away: 9 + 1 = 10, and 10 ÷ 0.4 = 25. Choosing 2 misplaces the decimal point in the gradient, dividing by 4 instead of by 0.4: 8 ÷ 4 = 2.
- (c) 60 minutes — Method: the median is the middle value once the data have been put in order of size, so the list must be sorted before any position is read. Working: in order the times are 50, 55, 60, 65, 90 minutes; there are 5 values, so the middle position is the third and the time sitting there is 60 minutes. Answer: 60 minutes. The distractors: 64 minutes comes from working out the mean, 320 ÷ 5, instead of the median; 70 minutes comes from taking the time halfway between the shortest and the longest, (50 + 90) ÷ 2; 40 minutes comes from working out the range, 90 − 50, which measures spread rather than centre.
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