Statistics

How is the estimated mean of grouped data calculated?

• AMiddle value of the data
• BMidpoint × frequency, summed, ÷ total frequency✓
• CMost common class
• DLargest − smallest

Answer: Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency.

The median is best described as:

• AThe mean of all values
• BThe middle value when ordered✓
• CThe most common value
• DThe spread of the data

Answer: The median is the middle value of an ordered data set.

Which average is most affected by an extreme outlier?

• AMode
• BMedian
• CMean✓
• DRange

Answer: The mean uses every value, so a very large or small outlier pulls it noticeably.

A scatter graph shows points going up from bottom-left to top-right. This is:

• ANo correlation
• BPositive correlation✓
• CNegative correlation
• DA pie chart

Answer: As one variable increases the other increases — that is positive correlation.

The interquartile range (IQR) is calculated as:

• AUpper quartile − lower quartile✓
• BMean − mode
• CMedian × 2
• DMaximum − minimum

Answer: IQR = upper quartile (Q3) − lower quartile (Q1); it measures the spread of the middle 50%.

In a histogram the bar for the class 20 < t ≤ 40 has a frequency density of 1.8. Work out the frequency for that class.

• A1.8
• B0.09
• C72
• D36✓

Answer: Frequency density = frequency ÷ class width, so frequency = frequency density × class width. The class width is 40 − 20 = 20, giving 1.8 × 20 = 36.

A histogram has a bar covering 0 < t ≤ 10 with a frequency density of 3.5, and a bar covering 10 < t ≤ 30 with a frequency density of 1.25. Work out the total frequency for 0 < t ≤ 30.

• A47.5
• B60✓
• C72.5
• D4.75

Answer: The classes have different widths, so each frequency must be found separately: 10 × 3.5 = 35 and 20 × 1.25 = 25, giving a total of 60. Frequency densities cannot simply be added.

A cumulative frequency graph shows the finishing times of 80 runners. Reading from the graph, a cumulative frequency of 20 gives 34 minutes, 40 gives 39 minutes and 60 gives 45 minutes. Work out the interquartile range.

• A11 minutes✓
• B5 minutes
• C6 minutes
• D39 minutes

Answer: With 80 values the lower quartile is read at a cumulative frequency of 20 and the upper quartile at 60, giving 34 and 45 minutes. The interquartile range is 45 − 34 = 11 minutes; the reading at 40 is the median.

A box plot has a minimum of 12, a lower quartile of 19, a median of 26, an upper quartile of 35 and a maximum of 58. Work out the range and the interquartile range.

• ARange 46, interquartile range 23
• BRange 39, interquartile range 16
• CRange 46, interquartile range 16✓
• DRange 32, interquartile range 9

Answer: The range uses the two whisker ends: 58 − 12 = 46. The interquartile range uses the two ends of the box: 35 − 19 = 16, which ignores the extreme values at each end.

For a set of data the lower quartile is 24 and the upper quartile is 40. A value counts as an outlier if it lies more than 1.5 × the interquartile range above the upper quartile. Work out the value at which that upper outlier boundary sits.

• A56
• B64✓
• C76
• D100

Answer: The interquartile range is 40 − 24 = 16, so 1.5 × 16 = 24. Adding that to the upper quartile gives 40 + 24 = 64, and anything above 64 is an outlier.