Comparing measurements

Activities to try

1. 1

   Watch the averages diverge: move a value far out

   Why it works: The mean is influenced by every value, so a single extreme value shifts it a lot; the median, based on order/position, is robust and barely moves. 'Robust' becomes concrete when you see the two measures respond differently to the same far-out value.

   1. Start with a small set of values on a stated scale (for example reaction times in whole seconds, or scores out of 40) clustered around 4-5, and compute the mean, median, mode and range by hand.
   2. Record all four measures, then take ONE of those values and move it far out to a value the scale allows (for example 30), keeping the number of data points the same.
   3. Recompute the mean and median with the far-out value, and compare each to its value before.
   4. State in words which measure moved a lot and which barely moved, and explain why the mean was dragged while the median held.
   5. If a draggable dot plot with mean, median, mode and range read out live is available, drag one point far from the cluster and watch the mean line chase it while the median stays put or moves at most to a neighbouring value in the cluster (for example, if the dragged point crosses from below the middle to above it) — this makes the differential effect visible in real time.
2. 2

   Same mean, different spread: why centre is not enough

   Why it works: An average alone does not describe a distribution. Two sets with equal means can differ widely in variability, so a fair comparison reports a measure of centre AND a measure of spread.

   1. Take two data sets (for example two classes' test scores) that have the same mean but clearly different spreads.
   2. Compute the mean and the range (highest minus lowest, a single number) for each set.
   3. Decide whether 'same mean' means the classes performed identically, and justify the answer using the ranges.
   4. Write a conclusion in context that mentions BOTH centre and spread, e.g. 'Both classes scored about the same on average, but Class A was far more consistent.'
   5. If side-by-side dot plots with adjustable spreads at a fixed mean are available, line the two plots up and compare how wide each is while the mean stays equal.
3. 3

   Which average would you report, and why?

   Why it works: The appropriate average depends on the data: the median represents a typical value better for skewed data or data with an outlier; the mode suits most-common / categorical questions; the mean suits roughly symmetric numerical data. Learners' informal notions of 'average' (most frequent, reasonable value, midpoint, algorithm) are each true only in some circumstances.

   1. Given a skewed data set (or one with an outlier), compute mean, median and mode and note where each falls.
   2. Choose which average best represents a typical value and write a one-sentence justification tied to the data's shape.
   3. For a most-common / categorical question, decide whether the mode is the right choice and say why.
   4. Check a tempting claim such as 'the mean always sits in the middle of the range' against your skewed data and explain whether it holds.