ITPEC IP April 2023 Question 1

Histogram (d), the U shape — variance measures how far the data lie from the mean, and (d) puts most of its data at both ends, farthest from its centre.

Variance is the mean of the squared distances from the mean, so values far from the centre weigh heavily. Numbering the data classes 0 to 14 and reading the bar heights gives:

In (d), 18 of the 25 values sit four or more classes away from the mean.

Answer (d)

Why not others:

• •(a) — its tail reaches far to the right, but most of the data crowd into the first few classes, close to their own mean
• •(b) — spread evenly over the range, so the middle is as full as the ends and the average squared distance is smaller than in (d)
• •(c) — bell-shaped: the data cluster around the centre, giving the smallest variance of the four

Key rule: Over the same range, variance is largest when the data pile up at both extremes and smallest when they cluster in the middle; judge by distance from the mean, not by the height of the tallest bar.

Basic Theory · AI-written, may contain errors