S1 2.2, S1 2.3 - Location and spread

Mean median and mode for data types

If every delivery takes ten minutes longer, its mean increases by ten minutes. Does its variability increase too? We will separate where a distribution lies from how widely its observations vary.

Discrete data take separated possible values, such as numbers of parcels. Continuous data measure quantities such as time or mass. Either can be given as individual observations, an exact-value frequency table, or a grouped table. Grouping loses the positions of observations within each class.

The mean uses every observation and is sensitive to a few extreme values. The median depends on position and is usually less affected by extremes. A mode can describe the most common category, such as a colour, even when a numerical mean would have no meaning.

Use cumulative frequencies to locate a median, rather than treating each row as one observation. For grouped continuous data, identify the median class first; interpolation in the next part will estimate a value within it. With unequal class widths, the modal class is the class with greatest frequency density, rather than necessarily greatest frequency.

Change only how precisely the values are known: replace exact values by classes with the same midpoints and frequencies.

The choice of summary depends on the question being asked, not on which number is quickest to calculate.

Estimate a mean from new grouped measurements.

Range interpercentile ranges and interpolation

A measure of spread says how far apart observations are; a percentile range can focus on a chosen central portion instead of the extremes.

The range is maximum minus minimum. The percentile $P_p$ marks the position below which approximately $p\%$ of observations lie. Thus $Q_1=P_{25}$, the median is $P_{50}$, and $Q_3=P_{75}$. The interquartile range $Q_3-Q_1$ spans the middle $50\%$; $P_{90}-P_{10}$ spans the middle $80\%$. A large range with a small IQR suggests that the extremes are much more separated than the central half.

For grouped continuous data, use target cumulative frequency $r=pn/100$. If it falls within a class, linear interpolation assumes a uniform spread within that class. Covering a fraction of the class frequency therefore covers the same fraction of its width.

The subtraction $r-c$ counts how many observations into the selected class you need to go. The denominator is that class's frequency, not its cumulative frequency. If a target falls exactly at a class boundary, either adjacent-class calculation gives that boundary.

Use the following grouped waiting times:

[DIAGRAM: asset_name: Grouped quartile interpolation; asset_slug: grouped-quartile-interpolation; caption: Going two observations into a class of twelve means going two twelfths of the way across its width.; description: The waiting-time class ten to twenty minutes contains twelve observations, with eight observations before it. A target cumulative frequency of ten lies two of those twelve observations into the class, at eleven and two thirds minutes. The first two twelfths of the interval are shaded separately.]

Change only the requested percentile, from $25\%$ to $75\%$. The new position lies in a different class, so its width and preceding cumulative frequency must change too.

The same process gives $P_{10}\approx0+(4/8)10=5$ and $P_{90}\approx20+((36-20)/20)20=36$. Their difference, $31$ minutes, spans the estimated central $80\%$. These class boundaries do not tell us the exact sample minimum or maximum, so $40$ is not necessarily the actual range.

You can also interpolate between known percentile positions. For example, moving halfway from $Q_1$ to the median corresponds to moving halfway from cumulative proportion $0.25$ to $0.50$, giving $0.375$, provided the same uniform-spread assumption is made for that interval. This is an estimate, not information supplied by the box plot alone.