Handling Data and Statistics

A driver records the number of parcels delivered on 7 days: 42, 38, 51, 45, 38, 49, 40. What is the median number of parcels?

• A38
• B42✓
• C43.3
• D45

Answer: Put the values in order: 38, 38, 40, 42, 45, 49, 51. With 7 values the median is the 4th one, which is 42. Taking the middle of the unsorted list gives 45, 38 is the mode and 43.3 is the mean.

A shoe shop sold these sizes on Saturday: 6, 8, 7, 9, 8, 5, 8, 7, 10, 7, 8. The manager wants to reorder the size that sells most often. Which average should she use, and what is its value?

• AThe mean, which is size 7.5
• BThe mode, which is size 8✓
• CThe mode, which is size 7
• DThe range, which is 5 sizes

Answer: The size that sells most often is the mode. Size 8 appears 4 times and size 7 appears 3 times, so the mode is size 8. The mean (83 ÷ 11 ≈ 7.5) is not a size anyone buys, and the range measures spread, not the most popular size.

Eight employees' journey times to work, in minutes, are: 12, 35, 20, 18, 27, 40, 15, 22. What is the median journey time?

• A21 minutes✓
• B22.5 minutes
• C20 minutes
• D23.6 minutes

Answer: In order: 12, 15, 18, 20, 22, 27, 35, 40. With 8 values the median is halfway between the 4th and 5th values: (20 + 22) ÷ 2 = 21 minutes. Using the unsorted middle pair (18 and 27) gives 22.5, and 23.6 is roughly the mean.

In a staff survey, 20 part-time staff have a mean age of 35 and 30 full-time staff have a mean age of 40. What is the mean age of all 50 staff?

• A38✓
• B37.5
• C37
• D75

Answer: Total age of part-time staff: 20 × 35 = 700. Total for full-time staff: 30 × 40 = 1,200. Mean of all staff: (700 + 1,200) ÷ 50 = 1,900 ÷ 50 = 38. Averaging the two means (37.5) ignores the different group sizes.

A sales rep needs a mean of at least 10 sales a week over 4 weeks to earn a bonus. In the first 3 weeks she made 7, 9 and 11 sales. What is the least number of sales she needs in week 4?

• A13✓
• B10
• C40
• D9

Answer: A mean of 10 over 4 weeks needs a total of 4 × 10 = 40 sales. So far she has 7 + 9 + 11 = 27. She needs at least 40 - 27 = 13 in week 4. The target mean (10) is not enough because the earlier weeks were below it on average.

Two bakers record daily loaf sales over 5 days. Baker A sold 40, 42, 41, 43 and 44. Baker B sold 20, 65, 30, 60 and 35. Which statement is correct?

• ABaker A sells more on average, and Baker A has the larger range
• BBaker B sells more on average, and Baker B is more consistent
• CThey have the same mean, but Baker B's sales are more consistent
• DThey have the same mean, but Baker A's sales are more consistent✓

Answer: Both totals are 210, so both means are 210 ÷ 5 = 42. Baker A's range is 44 - 40 = 4 and Baker B's range is 65 - 20 = 45. A smaller range means more consistent sales, so Baker A is more consistent.

If the median of 5 numbers is 15, which statement is true?

• AThe mean of the five numbers must also be 15
• BEvery one of the five numbers must be 15 or more
• CThe range of the five numbers must be 15
• DIn order, the third of the five numbers is 15✓

Answer: The median is the middle value once the numbers are in order. With five numbers, that is the third one, so it must be 15. The mean, the range and the other values can be anything.

A works social club sells 250 raffle tickets and Jodie buys 10 of them. What is the probability that Jodie holds the winning ticket?

• A0.4
• B0.04✓
• C0.1
• D0.025

Answer: Jodie holds 10 of the 250 tickets, and 10 ÷ 250 = 0.04.

A machine produces 3 faulty items in every 200 made. What is the probability that an item taken at random is not faulty?

• A0.015
• B0.97
• C0.15
• D0.985✓

Answer: 197 of every 200 items are free of faults, and 197 ÷ 200 = 0.985.

The probability that a particular bus is late is 0.15. Out of 60 journeys, how many would you expect to be late?

• A9✓
• B51
• C15
• D0.9

Answer: The expected number is the probability multiplied by the number of journeys: 0.15 × 60 = 9 journeys.

A quality check shows that 4% of light bulbs fail. In a delivery of 1,500 bulbs, how many would you expect to fail?

• A600
• B6
• C60✓
• D375

Answer: 4% of 1,500 is 0.04 × 1,500 = 60 bulbs.

In a prize draw the probability of winning a meal voucher is 0.1 and the probability of winning a cinema ticket is 0.05. One ticket cannot win both prizes. What is the probability of winning one of these two prizes?

• A0.005
• B0.85
• C0.5
• D0.15✓

Answer: The two outcomes cannot both happen, so the probabilities are added: 0.1 + 0.05 = 0.15.

A vending machine gives a cold drink with a probability of 0.4 and a hot drink with a probability of 0.35. The only other thing it gives is a snack. What is the probability of getting a snack?

• A0.75
• B0.05
• C0.25✓
• D0.14

Answer: All the probabilities must add up to 1, so the snack probability is 1 − 0.4 − 0.35 = 0.25.

A shift survey covers 80 staff. Of the 45 who work nights, 24 drive to work. Of the 35 who work days, 16 drive to work. What is the probability that a member of staff chosen at random both works nights and drives to work?

• A0.533
• B0.6
• C0.3✓
• D0.5

Answer: 24 of the 80 staff surveyed both work nights and drive, and 24 ÷ 80 = 0.3.

A college has 200 learners. 120 study maths, of whom 90 passed. The other 80 study English, of whom 60 passed. What is the probability that a learner chosen at random passed?

• A0.75✓
• B0.6
• C0.45
• D0.25

Answer: Altogether 90 + 60 = 150 learners passed out of 200, and 150 ÷ 200 = 0.75.

A café runs a promotion. A customer tosses a coin and rolls a fair six-sided dice, and wins a free drink only if they get a head and a six. What is the probability that a customer wins?

• A1/12✓
• B1/8
• C1/6
• D7/12

Answer: The two results do not affect each other, so the probabilities are multiplied: 1/2 × 1/6 = 1/12.

The probability that a delivery van breaks down on any given day is 0.1, and the days do not affect each other. What is the probability that it breaks down on two particular days in a row?

• A0.2
• B0.02
• C0.01✓
• D0.05

Answer: For both days to happen the probabilities are multiplied: 0.1 × 0.1 = 0.01.

A supermarket records what 60 customers spend. £0 up to £10: 8 customers. £10 up to £20: 21 customers. £20 up to £30: 19 customers. £30 up to £40: 12 customers. Which is the modal class?

• A£20 up to £30
• B£30 up to £40
• C£0 up to £10
• D£10 up to £20✓

Answer: The modal class is the class with the highest frequency, and the largest frequency here is 21 customers, in the class £10 up to £20.

Twenty staff record how long they take to get to work. 0 up to 10 minutes: 4 staff. 10 up to 20 minutes: 6 staff. 20 up to 30 minutes: 7 staff. 30 up to 40 minutes: 3 staff. Estimate the mean travel time.

• A14.5 minutes
• B19.5 minutes✓
• C24.5 minutes
• D97.5 minutes

Answer: Each class is represented by its midpoint: (4 × 5) + (6 × 15) + (7 × 25) + (3 × 35) = 390, and 390 ÷ 20 staff = 19.5 minutes.

Fifty households record their weekly shopping bill. £20 up to £40: 9 households. £40 up to £60: 16 households. £60 up to £80: 15 households. £80 up to £100: 10 households. Estimate the mean weekly bill.

• A£50.40
• B£60.40✓
• C£70.40
• D£755.00

Answer: Using the midpoint of each class: (9 × £30) + (16 × £50) + (15 × £70) + (10 × £90) = £3,020, and £3,020 ÷ 50 households = £60.40.

A manager records how long her staff take to travel to work in a grouped frequency table. To estimate the mean journey time, which value should she use to stand for the class '30 up to 40 minutes'?

• A35 minutes✓
• B30 minutes
• C40 minutes
• D70 minutes

Answer: Each class is represented by its midpoint, and halfway between 30 and 40 is 35 minutes.

A delivery firm works out a mean delivery time from a grouped frequency table. Why can that mean only ever be an estimate?

• AThe frequencies in a grouped table are rounded to the nearest ten before they are used
• BThe individual values are not recorded, so the midpoint of each class stands in for them✓
• CA grouped frequency table always leaves out the highest and the lowest values collected
• DThe classes in a grouped frequency table are always of different widths

Answer: A grouped table shows only how many values fall in each class, not the values themselves, so the midpoint has to stand in for them and the mean that results is an estimate.

A café records the highest temperature each day and how many bowls of hot soup it sells. On hotter days it sells fewer bowls. What type of correlation does a scatter diagram of this data show?

• APositive correlation: both variables rise together
• BNegative correlation: one variable falls as the other rises✓
• CZero correlation: the points show no clear pattern
• DPositive correlation: one variable falls as the other rises

Answer: When one variable goes up as the other goes down, the points slope down from left to right. That is negative correlation. Positive correlation is when both rise together, and zero correlation means no clear pattern.

A scatter diagram shows the age of used cars against their price. The line of best fit passes through (2 years, £12,000) and (8 years, £3,000). Use the line to estimate the price of a 5-year-old car.

• A£10,500
• B£7,500✓
• C£6,000
• D£4,500

Answer: Between 2 and 8 years, the price falls by £9,000 over 6 years, which is £1,500 a year. From 2 to 5 years is 3 years, so the price falls by 3 × £1,500 = £4,500: £12,000 - £4,500 = £7,500. 5 years is halfway along, so the estimate is halfway between the prices too. £4,500 is the fall, not the price.

A scatter diagram shows strong positive correlation between daily ice-cream sales and the number of sunburn cases at a beach. Which conclusion is most sensible?

• AGetting sunburnt makes people want to buy more ice cream
• BThe diagram shows there is no link between the two sets
• CA third factor, such as hot sunny weather, may explain both✓
• DEating ice cream makes people more likely to get sunburnt

Answer: Correlation shows the two sets of data rise together, but it does not prove that one causes the other. Sunny weather increases both ice-cream sales and sunburn, so it is the likely explanation. A strong correlation is clearly a link, so there is a relationship to explain.

A tutor records hours of revision and test scores for six learners: (2, 35), (4, 48), (5, 55), (7, 66), (8, 74) and (3, 90). On a scatter diagram, which point is most likely to be an outlier?

• A(2, 35)
• B(5, 55)
• C(3, 90)✓
• D(8, 74)

Answer: Five of the points show a clear pattern: more revision goes with a higher score, rising by roughly 6 to 7 marks per hour. A learner who revised for 3 hours would be expected to score about 40, so a score of 90 is far from the pattern and is the outlier.

A scatter diagram of adults' heights (between 1.5 m and 1.9 m) and shoe sizes shows positive correlation. Why would using the line of best fit to predict the shoe size of someone 2.5 m tall be unreliable?

• AA line of best fit is meant to join up each plotted point
• BA line of best fit should always pass through the origin
• CShoe sizes are whole numbers, so a line can't be drawn
• DIt goes well beyond the range of the data collected✓

Answer: The data only covers heights from 1.5 m to 1.9 m. Using the line far outside that range (extrapolation) assumes the pattern carries on, which may not be true. A line of best fit does not have to pass through the origin or through each point.

A gym plots each member's weekly exercise hours against their resting heart rate. The points slope downwards from left to right and lie close to a straight line. Which description fits?

• AWeak positive correlation
• BStrong negative correlation✓
• CWeak negative correlation
• DStrong positive correlation

Answer: Points sloping down from left to right show negative correlation: more exercise goes with a lower resting heart rate. Because the points lie close to a straight line, the correlation is strong. Weak correlation would show points widely scattered around the line.

On a scatter diagram of outdoor temperature (°C) against electricity used (kWh), the line of best fit passes through (0, 50) and (20, 10). Use the line to estimate the temperature on a day when 30 kWh is used.

• A10°C✓
• B15°C
• C5°C
• D20°C

Answer: The line falls by 40 kWh over 20°C, which is 2 kWh for each degree. To fall from 50 kWh to 30 kWh is a drop of 20 kWh, which takes 20 ÷ 2 = 10 degrees, so the temperature is 10°C. 30 kWh is halfway between 50 and 10, so the temperature is halfway between 0 and 20.

In a survey of 120 commuters, 70 drive and 50 take the train. Of the drivers, 45 work full-time. Of the train users, 20 work full-time. One commuter is chosen at random. What is the probability, as a percentage, that they take the train and work part-time?

• A60%
• B25%✓
• C16.7%
• D41.7%

Answer: Train users who work part-time: 50 - 20 = 30. The probability is 30 out of all 120 commuters: 30 ÷ 120 = 0.25 = 25%. 60% is 30 out of the 50 train users, which answers a different question.

A spinner lands on red with a probability of 0.35 and on blue with a probability of 0.4. Otherwise it lands on green. What is the probability of green, as a fraction in its simplest form?

• A1/5
• B1/25
• C1/4✓
• D3/4

Answer: Probabilities add up to 1, so P(green) = 1 - 0.35 - 0.4 = 0.25. As a fraction, 0.25 = 25/100 = 1/4. 3/4 is the probability of red or blue, and 1/25 misreads 0.25 as 1 out of 25.