would be classed as being in the interval. For example, had we recorded reaction times to
the nearest thousandth of a second, rather than to the nearest hundredth, the interval 35–39
would include all values between 34.5 and 39.5 because values falling between those
points would be rounded up or down into that interval. (People often become terribly wor-
ried about what we would do if a person had a score of exactly 39.50000000 and therefore
sat right on the breakpoint between two intervals. Don’t worry about it. First, it doesn’t
happen very often. Second, you can always flip a coin. Third, there are many more impor-
tant things to worry about. Just make up an arbitrary rule of what you will do in those situ-
ations, and then stick to it. This is one of those non-issues that make people think the study
of statistics is confusing, boring, or both.)
Themidpointslisted in Table 2.3 are the averages of the upper and lower limits and
are presented for convenience. When we plot the data, we often plot the points as if they all
fell at the midpoints of their respective intervals.
Table 2.3 also lists the frequencies with which scores fell in each interval. For example,
there were seven reaction times between 35/100 and 39/100 of a second. The distribution
in Table 2.3 is shown as a histogram in Figure 2.2.
People often ask about the optimal number of intervals to use when grouping data. Al-
though there is no right answer to this question, somewhere around 10 intervals is usually
reasonable.^2 In this example I used 19 intervals because the numbers naturally broke that
way and because I had a lot of observations. In general and when practical, it is best to use
natural breaks in the number system (e.g., 0–9, 10–19,... or 100–119, 120–139) rather
than to break up the range into exactly 10 arbitrarily defined intervals. However, if another
kind of limit makes the data more interpretable, then use those limits. Remember that you
are trying to make the data meaningful—don’t try to follow a rigid set of rules made up by
someone who has never seen your problem.20 Chapter 2 Describing and Exploring Data
40 60 80 100 120
RxTime2030401050FrequencyReaction TimesFigure 2.2 Grouped histogram of reaction times(^2) One interesting scheme for choosing an optimal number of intervals is to set it equal to the integer closest to,
where Nis the number of observations. Applying that suggestion here would leave us with
intervals, which is close to the 19 that I actually used. Other rules are attributable to Sturges, Scott,
and Freeman-Diaconis.
17.32= 17
1 N 1 N= 1300 =
midpoints