Descriptive statistics
A measured value without its spread is not a measurement: it is a number. All the descriptive statistics in this unit exist so that you can write the “plus or minus” that follows the value.
01Why a single number is not enough
Measure the same resistance ten times with the same instrument and you do not get the same number ten times. That scatter is not a flaw in the procedure: it is information, and throwing it away amounts to inventing a precision you do not have.
Descriptive statistics summarizes a data set with two questions: around what value the data lie and how spread out they are. Everything else is detail.
| Measure | What it tells you | When to use it |
|---|---|---|
| Mean \( \bar{x} \) | The arithmetic average | Data without outliers |
| Median | The middle value when the data are sorted | With outliers: it does not get dragged along |
| Mode | The most frequent value | Discrete or categorical data |
| Standard deviation \( s \) | The typical spread | The standard measure of uncertainty |
| Interquartile range | The width of the central 50 % | With outliers, like the median |
That \( \sqrt{n} \) has a hard practical consequence: to improve the uncertainty tenfold you have to measure a hundred times more. That is why averaging has diminishing returns, and at some point it pays to improve the instrument instead of continuing to measure.
02See the shape, not just the numbers
Two data sets can have the same mean and the same standard deviation and still be completely different. That is why a histogram is worth more than any summary: it shows the shape.
Five batches with different histories. Watch how the mean and the median drift apart when there are outliers, and what happens with a batch that was screened before it reached you.
A manufacturer that produces resistors and sorts them by tolerance sells the ones that fell within 1 % as 1 % components, and the rest as 5 %. The result: the “5 %” batch has a hole in the middle, and its histogram is bimodal. The mean still gives the nominal value and the standard deviation looks reasonable, but the shape looks nothing like a bell curve. It is the classic example of why looking at the histogram is not optional.
03Quartiles and outliers
When the data are sorted, the quartiles split them into four equal groups. The first leaves 25 % below it, the second is the median and the third leaves 75 % below it. The distance between the first and the third —the interquartile range— measures the spread without a single absurd data point ruining it.
Discarding a data point just because it is inconvenient is the worst sin in the lab. An outlier can be a typo, a badly connected instrument or a different part that slipped into the batch —all legitimate reasons to exclude it, as long as you document it— but it can also be the only data point that shows the real problem. The rule is to look at what produced it before deciding.
04In the lab
Measure fifty resistors of the same nominal value and tolerance. Build the histogram, compute the mean, standard deviation and quartiles, and check whether the measured spread is consistent with the stated tolerance.
Measure the same resistance thirty times without touching anything. Compute the standard deviation of the measurements and the uncertainty of the average. Repeat with ten measurements and verify the \( \sqrt{n} \) relationship between the two uncertainties.
Measure a batch of 5 % tolerance resistors and build the histogram with narrow classes. Check whether it has a bell shape or whether the center is missing, and explain what each result would reveal.
05Common mistakes
- Reporting an average without its spread. A value without its “plus or minus” is not a measurement.
- Using the mean with outliers. A single one shifts it; the median does not budge.
- Dividing by \( n \) instead of \( n-1 \). It underestimates the spread, and with few data points by quite a bit.
- Confusing the standard deviation of the data with that of the average. The latter is \( s/\sqrt{n} \) and is much smaller.
- Looking only at the numbers and not at the histogram. Two very different data sets can have the same summaries.
- Deleting a data point because it is inconvenient. First find out what produced it.
06Self-assessment
Why does the sample standard deviation divide by \( n-1 \)?
Because the mean was computed from the same data, so one degree of freedom is lost. Dividing by \( n \) underestimates the spread.
How much does the uncertainty improve if you measure a hundred times more?
Tenfold: it falls with the square root of \( n \). That is why averaging has diminishing returns.
When is the median preferable to the mean?
When there are outliers: the median does not get dragged along by them.
What is the interquartile range?
The distance between the first and the third quartile: the width of the central 50 % of the data.
Why can a 5 % batch be bimodal?
Because the parts from the center were removed to be sold as 1 % parts. The histogram ends up with a hole in the middle.
What do you do with an outlier?
Investigate what produced it. It can be excluded if there is an identified cause, and documenting it.
07Further reading
- Douglas C. Montgomery and George C. Runger. Applied Statistics and Probability for Engineers. 7th ed. (Spanish edition, “Probabilidad y estadística aplicadas a la ingeniería”), Wiley, 2018. The standard reference for the course. Chapters 1 and 6 cover this unit with engineering data.
- John W. Tukey. Exploratory Data Analysis. Addison-Wesley, 1977. The source of the box plot and the 1.5 · IQR criterion, with the argument for why you have to look at the data before summarizing it.
- JCGM. Evaluation of measurement data — Guide to the expression of uncertainty in measurement. JCGM 100:2008 (Spanish edition published by the BIPM). The standard that defines how a measurement is reported. Type A and Type B uncertainties come from here.