Question 3.26: Use a tally chart to determine the frequency of events for t...
Use a tally chart to determine the frequency of events for the data given above on the assembly-line task.
\begin{array}{llllllllll}1.1 & 1.0 & 0.6 & 1.1 & 0.9 & 1.1 & 0.8 & 0.9 & 1.2 & 0.7 \\1.0 & 1.5 & 0.9 & 1.4 & 1.0 & 0.9 & 1.1 & 1.0 & 1.0 & 1.1 \\0.8 & 0.9 & 1.2 & 0.7 & 0.6 & 1.2 & 0.9 & 0.8 & 0.7 & 1.0 \\1.0 & 1.2 & 1.0 & 1.0 & 1.1 & 1.4 & 0.7 & 1.1 & 0.9 & 0.9 \\0.8 & 1.1 & 1.0 & 1.0 & 1.3 & 0.5 & 0.8 & 1.3 & 1.3 & 0.8\end{array}
| Time (hours) | Tally | Frequency |
| 0.5 | 1 | 1 |
| 0.6 | 1 1 | 2 |
| 0.7 | 1 1 1 1 | 3 |
| 0.8 | \sout{1111 \ 1} | 4 |
| Time (hours) | Tally | Frequency |
| 0.9 | \sout{1111 \ 1}\ 1\ 1 | 8 |
| 1.0 | \sout{1111 \ 1111 \ 1} | 1 1 |
| 1.1 | \sout{1111 \ 1}\ 1 \ 1 | 8 |
| 1.2 | 1 1 1 1 | 4 |
| 1.3 | 1 1 1 | 3 |
| 1.4 | 1 1 | 2 |
| 1.5 | 1 | 1 |
| Total | 50 |
We now have a full numerical representation of the frequency of events. For example, eight people completed the task in 1.1 hours, or the time 1.1 hours has a frequency of 8. We will be using the above information later on, when we consider measures of central tendency.
The time in hours given in the above data are simply numbers. When data appears in a form where it can be individually counted we say that it is discrete data. It goes up or down in countable steps. Thus the numbers 1.2, 3.4, 8.6, 9, 11.1 and 13.0 are said to be discrete. If, however, data is obtained by measurement, e.g. the heights of a group of people, then we say that this data is continuous. When dealing with continuous data, we tend to quote its limits, i.e. the limit of accuracy with which we take the measurements. For example, a person may be 174 ± 0.5 cm in height. When dealing numerically with continuous data or a large amount of discrete data, it is often useful to group this data into classes or categories. We can then find out the numbers (frequency) of items within each group.
The following table shows the height of 200 adults grouped into 10 classes.
| Height (cm) | Frequency |
| 150 – 154 | 4 |
| 155 – 159 | 9 |
| 160 – 164 | 15 |
| 165 – 169 | 21 |
| 170 – 174 | 32 |
| 175 – 179 | 45 |
| 180 – 184 | 41 |
| 185 – 189 | 22 |
| 190 – 194 | 9 |
| 195 – 199 | 2 |
| Total | 200 |
The main advantage of grouping is that it produces a clear overall picture of the frequency distribution. In the table, the first class interval is 150–154. The end number 150 is known as the lower limit of the class interval; the number 154 is the upper limit. The heights have been measured to the nearest centimetre. That means within ± 0.5 cm. Therefore, in effect, the first class interval includes all heights between the range 149.5–154.5 cm. These numbers are known as the lower and upper class boundaries, respectively. The class width is always taken as the difference between the lower and upper class boundaries, not the upper and lower limits of the class interval.