(I hope the question in in scope, please see my question on Meta about that)
My 15 yo son (2nde in France, this is the first year of the equivalent of a High School) is going through basic statistics. One of the exercises in his book made me wonder about why some simplifications are made in an exercise.
While I would love to be corrected and learn something, the core of my question is rather whether this exercise is
- simplified because it will help to understand something later (and one can live with the simplification)
- simplified because this is the right thing to do (this is the moment where I would learn something about statistics)
- a bad exercise (and the problem is over)
The problem(1) is :
In 2016 women were asked about their age at the time they got married. The table below contains their answers:
$$ \begin{array} {|r|r|}\hline Age & 20 \le A \lt 30 & 30 \le A \lt 40 & 40 \le A \lt 50 & 50 \le A \lt 60 & 60 \le A \lt 70 \\ \hline Amount & 10 & 35 & 15 & 2 & 1 \\ \hline \end{array} $$
Calculate the mean age of marriage for these women. Calculate the standard deviation of this series.
The answer then proceeds with
In order to determine the mean, we assume that a person who is between 20 and 30 years old is 25 years old.
Then the problems is solved as expected for a series.
The assumption is, I believe, wrong, it depends on the distribution within the range. There are no reasons for the distribution to be symmetric.
A very similar problem was given later, where the range were apartment surfaces. For the range 0 to 20 m2, I expect that I should have stated that the average is 10 sqm, which makes even less sense as there obvious limits on the lower size of an apartment and the distribution will be heavily skewed to the right.
My problem with these exercises is that they make an artificial issue (= having to decide on the value to use in the ranges) while it is easy to have exercises such as "satisfaction level from 1 to 10 over a month, calculate the average & deviation" or "high jump - which athletes are more consistent in their results? (some would have low and high results, some more around the average for instance)).
My core questions are:
- Was there in the exercise I quoted a reason to do the assumption?
- Is this a good assumption?
- In what will it help later in their curriculum?
(1) Déclic 2nde Math, Programme 2019, Hachette. Ex. 3 p. 311