## Frequency

### Ungrouped Data

Ungrouped data is usually small amounts of data where individual values are listed. e.g. 23, 45, 67, 89

If many of these values are the same the data can be place in a frequency table. These tables can be horizontal or vertical.

e.g. Test results (out of 10) of twenty people: 3, 5, 8, 6, 3, 7, 7, 8, 5, 6, 4, 3, 6, 8, 9, 1, 4, 6, 2, 10

 Result (x) 0 1 2 3 4 5 6 7 8 9 10 Frequency (f) 0 1 1 3 2 2 4 2 3 1 1

This data, called a frequency distribution, can then be shown in a bar graph. ### Grouped Discrete Data

Grouped data is used when each value is likely to be different and values are placed in groups or classes.

e.g.Test results (%) of 30 people

 Score Frequency 0 − 25 3 26 − 50 8 51 − 75 12 76 − 100 7

This data is best shown in a bar graph. ### Grouped Continuous Data

A histogram is a graph that is used to show the information from a frequency distribution with grouped continuous data.

It is similar to a column graph and the area of each column is proportional to the frequency of the score that it represents.

e.g Ages of workers in a company.

 Ages Frequency 15 − 24 14 25 − 34 20 35 − 44 24 45 − 54 18 55 − 64 10

Note

Because these are ages,
15 − 24 means greater or equal to 15 and up and including 24. If any of the intervals in a frequency distribution are of a different size, the height of the bar above it, must be changed accordingly.

e.g. If the bottom interval in the table above was 55 − 74, twice as wide as the others, then the bar would be halved in height to 5.

frequency polygon is created when the midpoints of the bars of a histogram are joined.

### Cumulative Frequency Graph

A useful type of graph for finding, median, quartiles and percentiles is the cumulative frequency graph.

The grouped frequency table shows the times, in minutes, of the first 100 finishers in a 42 km marathon race.

 Time (minutes)x Frequency (number of runners)(f) The frequency table on the left can produce the cumulative frequency table on the right. Each term is the sum of all of the frequencies before it. In this case it tells how many runners finished in less than the time given. Time (minutes)x Cumulative Frequency number < x 120 − 130 6 120 0 130 − 140 14 130 6 140 − 150 20 140 6 + 14 = 20 150 − 160 50 150 20 + 20 = 40 160 − 170 64 160 40 + 50 = 90 170 − 180 46 170 90 + 64 = 154 200 180 154 + 46 = 200

From the cumulative frequency table a cumulative frequency graph can be drawn. Each point is joined with a straight line. From the cumulative frequency graph, the percentiles can be found.

Median (50th percentile). As there are 200 runners, the median will be the 200 x 0.5 = 100th runner.
Draw a horizonal line from 100. This gives a median of 162 minutes.

Upper quartile (75th percentile). As there are 200 runners, the upper quartile will be the 200 x 0.75 = 150th runner.
Draw a horizonal line from 150. This gives a median of 169 minutes.

Lower quartile (25th percentile). As there are 200 runners, the lower quartile will be the 200 x 0.25 = 50th runner.
Draw a horizonal line from 50. This gives a median of 152 minutes.

The xth percentile As there are 200 runners, to find the xth percentile, multiply 200 by x and divide by 100, this will give where to draw the horizonal line.
Remember, if you find the 90th percentile, 90% of the data lies below this value.