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Statistics · Data presentation

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Types of data and grouped frequency

Qualitative vs quantitative data, and within quantitative the split between discrete and continuous; how continuous data is grouped into classes, and how to read off the class boundaries, class width and midpoint you need before you can draw a histogram or estimate an average.

Statistics · Data presentation

Types of data and grouped frequency

Qualitative vs quantitative data, and within quantitative the split between discrete and continuous; how continuous data is grouped into classes, and how to read off the class boundaries, class width and midpoint you need before you can draw a histogram or estimate an average.

Why it works

Before you can summarise or draw data you have to know what kind it is, because that decides which tools are allowed.

Qualitative vs quantitative. Quantitative data is numerical — heights, times, number of goals. Qualitative (or categorical) data describes a quality with no numerical value — eye colour, type of car, cardinal wind direction (N, SE, …). You can count how many fall in each category, but you can't take a mean of "north".

Discrete vs continuous. Quantitative data divides again:
  • Discrete data can only take particular separate values — usually whole-number
counts. Number of children, shoe size, goals scored. Between 22 goals and 33 goals there's nothing.
  • Continuous data can take any value in a range, limited only by the accuracy
of measurement. Height, mass, time, temperature. A time could be 12.312.3 s, 12.3412.34 s, 12.33712.337 s…

Grouping data into classes. With a lot of continuous data we collect values into classes (intervals). Three numbers matter for each class, and getting them right is what makes histograms and grouped averages work:
  • Class boundaries — the values where one class actually stops and the next
starts (so there's no gap).
  • Class width == upper boundary −- lower boundary.
  • Midpoint == lower boundary+upper boundary2\dfrac{\text{lower boundary} + \text{upper boundary}}{2} — used
to represent the class when estimating a mean.

When the class is written with inequalities, the boundaries are obvious. For 10≤t<2010 \le t < 20 the boundaries are 1010 and 2020, the width is 1010 and the midpoint is 1515.

The hidden-gap case. When continuous data is recorded to the nearest unit, the printed limits hide the real boundaries. Heights recorded to the nearest cm and grouped 150150–159159, 160160–169169 look as if they have a 11 cm gap, but a height of 159.7159.7 cm rounds to 160160. So the class 150150–159159 really covers 149.5149.5 up to 159.5159.5:
  • boundaries 149.5149.5 and 159.5159.5,
  • width 159.5−149.5=10159.5 - 149.5 = 10,
  • midpoint 149.5+159.52=154.5\dfrac{149.5 + 159.5}{2} = 154.5.
The boundaries sit halfway into the gap between the stated limits. Miss this and your class width (and so your frequency density) is wrong by one unit.

Discrete grouped data is treated more simply — for 00–99, 1010–1919 goals the "classes" are just blocks of values; for averages you use the midpoint of the stated limits (e.g. 0+92=4.5\tfrac{0+9}{2} = 4.5). Histograms, though, are for continuous data, so when a histogram is asked of discrete-looking groups you use continuous boundaries.