Which Average for the Delivery Times?
A courier depot records the delivery time, in minutes, for 40 parcels on one morning. The mean is 34.2 minutes and the median is 29.0 minutes. The minimum is 18 minutes and the maximum is 96 minutes; three parcels took over 80 minutes because of a road closure. The manager says the depot's target is a mean of 30 minutes and claims the depot is "close enough". A customer survey separately records satisfaction on a 1–5 scale, and most customers give 5, with a handful giving 1. The depot also reports an interquartile range of 11 minutes.
1. Explain why the mean exceeds the median here, and advise which measure better describes a typical delivery time for this data set.
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The mean is pulled upward by the three extreme values above 80 minutes. The mean uses the actual value of every observation, so a long tail on the right drags it up. The median depends only on position in the ordered list, so it is resistant to outliers and stays near the bulk of the data.
Here mean 34.2 > median 29.0, a gap of 5.2 minutes, which is the signature of right skew: a few very slow deliveries and a dense cluster of ordinary ones.
Advice: for describing a typical delivery, the median (29.0 minutes) is the better summary, because it is not distorted by the road-closure parcels. The mean is still the right statistic for the manager's purpose - total delivery minutes divided by parcels - but it should not be presented as "typical" without the median and the range alongside it.
2. The manager claims the depot is "close enough" to the 30-minute target. Assess this claim using the figures, and explain the role of the interquartile range.
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The claim is weak. The mean of 34.2 is 4.2 minutes above target, about 14% over. More importantly, the mean conceals the shape: the median of 29.0 is actually below target, so the typical parcel is delivered within target while a small number of very slow deliveries push the average over.
This matters for the decision. If the target is about the average, the depot fails. If it is about most customers, the depot broadly meets it, but the tail is a real service failure for those affected.
The interquartile range of 11 minutes describes the spread of the middle 50% of deliveries: the middle half of parcels arrive within an 11-minute band. That is fairly consistent service for the bulk. But the IQR deliberately ignores the top 25%, so it says nothing about the slow tail - the range (18 to 96, a span of 78 minutes) is what exposes that. The two measures answer different questions and should be reported together.
3. For the 1–5 satisfaction ratings, recommend a measure of average and a measure of spread, justifying both.
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Average: the mode, or equivalently the median, rather than the mean. The ratings are ordinal - the gap between 4 and 5 need not equal the gap between 1 and 2 - so arithmetic on them is questionable. The distribution is also heavily skewed, with most customers at 5 and a few at 1, so the mean would sit in a region few customers actually chose. The mode describes the most common response directly, which is what a satisfaction headline should say.
Spread: report the proportion giving each rating (a small frequency table or bar chart) rather than a single number. Because the scale is ordinal and the distribution is bimodal, no single dispersion statistic is fully adequate; the range (1 to 5) is true but uninformative. A useful summary is "X% gave 5, Y% gave 1 or 2", which captures both the dominant response and the dissatisfied minority.
If the survey is treated as interval data by convention, the mean and standard deviation may be quoted - but the full distribution should still be shown, because the mean alone would misrepresent a split response.