Processing, representing and analysing data

The table shows (data are illustrative, written for this paper):

t (minutes)	Frequency
0 < t (minutes) ≤ 10	5
10 < t (minutes) ≤ 20	12
20 < t (minutes) ≤ 30	18
30 < t (minutes) ≤ 40	10
40 < t (minutes) ≤ 60	5

Calculate an estimate of the mean.

Explain why your answer to (a) is an estimate.

Calculate the frequency density for each class, ready to draw a histogram, and state which class has the greatest frequency density.

Use linear interpolation to estimate the median.

Estimate the 90th percentile.

Eleven students' journey times in minutes are (data are illustrative, written for this paper): 12, 15, 17, 18, 20, 21, 23, 25, 28, 30, 52. Find the median and the interquartile range.

An outlier is a value more than 1.5 × IQR below Q1 or above Q3. Show whether there are any outliers.

The number of books read by five students in a term (data are illustrative, written for this paper): 4, 7, 9, 10, 15. Calculate the mean and the standard deviation. (Use σ = √(Σ(x − x̄)²/n).)

A distribution has mean 25.1, median 23.8 and standard deviation 11.6. Calculate Pearson's coefficient of skewness, 3(mean − median) ÷ standard deviation, and describe the skew.

The product moment correlation coefficient between hours of revision and test score is 0.86. Interpret this value and state why it does not prove that revision causes higher scores.

Two judges ranked six cakes in a competition. (data are illustrative, written for this paper)

Item	A	B	C	D	E	F
Rank 1	1	2	3	4	5	6
Rank 2	2	1	4	3	6	5

Calculate Spearman's rank correlation coefficient and interpret it.

The equation of the regression line of ice creams sold on temperature (°C) is y = 3.2x + 14 (data are illustrative, written for this paper). Estimate ice creams sold when temperature (°C) is 8.

Interpret the gradient of the regression line in context.

The data used had temperature (°C) values from 0 to 30. Explain why using the line to estimate ice creams sold when temperature (°C) is 90 may be unreliable.

Quarterly sales (£000) for two years (data are illustrative, written for this paper): 40, 55, 70, 35, 44, 59, 74, 39. Calculate the first two 4-point moving averages and describe the trend.

The price of a loaf of bread was £2.40 in 2020 and £2.88 in 2024. Calculate the index number for 2024 using 2020 as base year.

A price was £50 in year 1, £54 in year 2 and £59.40 in year 3. Calculate chain base index numbers for years 2 and 3.

For the last two Quarter 4 periods, actual sales were 62 and 66 and the trend values were 51.5 and 53.5 (data are illustrative, written for this paper). The trend line gives 57.0 for next year's Quarter 4. Estimate next year's Quarter 4 sales.

Two classes sat the same test. Group A: 18 people, mean 64.5. Group B: 12 people, mean 71.0. Calculate the mean of the combined group.

In a survey of 120 people, 27 chose cycling (data are illustrative, written for this paper). Calculate the angle for cycling on a pie chart.

The table shows (data are illustrative, written for this paper):

m (kg)	Frequency
40 < m (kg) ≤ 50	8
50 < m (kg) ≤ 55	15
55 < m (kg) ≤ 60	22
60 < m (kg) ≤ 70	18
70 < m (kg) ≤ 90	7

Calculate an estimate of the mean.

Explain why your answer to (a) is an estimate.

Calculate the frequency density for each class, ready to draw a histogram, and state which class has the greatest frequency density.

Use linear interpolation to estimate the median.

Estimate the 90th percentile.

Fifteen daily rainfall totals (mm) for a month are (data are illustrative, written for this paper): 31, 34, 36, 38, 40, 41, 42, 44, 45, 47, 49, 50, 53, 55, 78. Find the median and the interquartile range.

An outlier is a value more than 1.5 × IQR below Q1 or above Q3. Show whether there are any outliers.

Five scores in a quiz (data are illustrative, written for this paper): 12, 15, 18, 20, 25. Calculate the mean and the standard deviation. (Use σ = √(Σ(x − x̄)²/n).)

A distribution has mean 46.2, median 49.0 and standard deviation 8.4. Calculate Pearson's coefficient of skewness, 3(mean − median) ÷ standard deviation, and describe the skew.

Two teachers ranked seven pieces of coursework. (data are illustrative, written for this paper)

Item	A	B	C	D	E	F	G
Rank 1	1	2	3	4	5	6	7
Rank 2	1	3	2	5	4	7	6

Calculate Spearman's rank correlation coefficient and interpret it.

The equation of the regression line of number of electric cars (thousands) on years since 2015 is y = 4.5x + 12 (data are illustrative, written for this paper). Estimate number of electric cars (thousands) when years since 2015 is 6.

Interpret the gradient of the regression line in context.

The data used had years since 2015 values from 0 to 8. Explain why using the line to estimate number of electric cars (thousands) when years since 2015 is 24 may be unreliable.

A weighted index is calculated for a family's shopping (data are illustrative, written for this paper). Bread: index 110, weight 3; Milk: index 120, weight 2; Eggs: index 105, weight 5. Calculate the weighted index number.

A town has a population of 60,000 and there were 450 deaths last year (data are illustrative, written for this paper). Calculate the crude death rate per 1,000.

Death rates per 1,000 in Town A and a standard population (data are illustrative, written for this paper):

Age	Rate per 1,000	Standard population
0–39	1.2	50%
40–64	5	30%
65+	40	20%

Calculate the standardised death rate for Town A.

Explain why standardised rates are better than crude rates for comparing towns.

Compare the distributions of two classes' test marks: Class A median 54, IQR 12, range 40; Class B median 61, IQR 22, range 45.

Give three reasons why a pie chart would be a poor choice for showing how the temperature in a town changes over a year.

For the last two Quarter 4 periods, actual sales were 48 and 53 and the trend values were 41.25 and 43.75 (data are illustrative, written for this paper). The trend line gives 47.5 for next year's Quarter 4. Estimate next year's Quarter 4 sales.