I am trying to calculate the approximate mean age of people that participated in a survey. I have 4 age groups: $[0,15);\,[15,35);\,[35,55)$ and $[55,75).$
For each group age I have the number of people that participated. How should I calculate the approximate mean age of people.
Thank you in advance, Mat
If you only care about the mean, you can take the mean of each group and multiply by the frequency of each group to obtain an unbiased estimator for the mean of your total sample, since the expected value operator is a linear operator. This estimator however will have larger variance.
Suppose we have data as shown in the histogram below. (Of course there are no ages below $0,$ but I wanted the histogram bars to be of equal widths.)
Suppose we make the simplifying approximation that all ages in each of the $k = 4$ histogram bins are at the centers of their intervals. Then we get frequencies $3, 41, 51, 5$ (for $n = 100$ subjects altogether) 'located' at midpoints $7, 25, 45, 65.$
Then the sample mean can be estimated as $$\bar X \approx \frac{1}{n}\sum_{i = 1}^k f_im_i = 36.66.$$
The average was computed, using R statistical software as a calculator, as follows:
f = c(3, 41, 51, 5)
m = c(7, 25, 45, 65)
a = sum(f*m)/100; a
[1] 36.66
Perhaps somewhat less accurately, we can approximate the sample variance as follows:
$$S^2 \approx \frac{1}{n-1}\sum_{i=1}^k f_i(m_i - \bar X)^2 = 159.358.$$
v = sum(f*(m-a)^2)/99; v
[1] 159.358
Unless you have the original data, you can't know how accurately $\bar X$ and $S^2$ are actually estimated by these formulas. However, I simulated the heights in R, so we can check the true values. For my simulated data the approximate values happen to be quite accurate. [I suspect results in this example turned out to be a little better than usual--especially because we used only four intervals.]
set.seed(2020) # for reproducibility
x = round(rnorm(100, 35, 10))
summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
5.00 29.00 36.00 36.09 42.25 67.00
var(x)
[1] 124.4666
Note: (1) Some years ago when computation was more tedious than it is with modern software, it was common practice to use the formulas shown here to approximate $\bar X$ and $S^2$ for large samples. Results are usually pretty good if you use a dozen or so intervals.
(2) Some elementary statistics books (especially high-school AP statistics books) have formulas for estimating sample medians, other sample quantiles, estimating population modes from data summarized in the form of intervals and frequencies.
(3) See this related Q&A.
