Grouped Data

Estimation from Histograms

(Note: for guidance on how to draw histograms, see Presenting Data.)

Since histograms collate data, it may seem impossible to answer questions such like how many data points are greater than 9, unless 9 is a class boundary. We can, however, estimate the answers to these questions by assuming frequency is evenly distributed across an entire class. Here is how to do it:

Example: Approximately how many values are greater than 12 in this histogram?

Draw a line at 12 on the x axis. This will split the second block. Then, the area of the graph to the right of the line is our estimate. In this case, the second block now extends from 12 to 20, with a height of 2, so a frequency of 2\times 8=16 comes from the second block. The third block has a length of 10 and a height of 3, so gives 30 frequency. In total, there are 16+30=46 values larger than 12.

Frequency Polygon

A frequency polygon is another way to represent grouped data. It is a line graph joining the points with co-ordinates (midpoint of class, frequency).

Example:

The midpoints are 5,14,21,25,33, so we plot the points:

(5,9)

(14,15)

(21,17)

(25,9)

(33,4)

and connect them with straight lines.

Estimating the Mean and Standard Deviation from a Histogram

Previously, when we used frequency tables to find the mean and standard deviation, we looked at x, fx and fx^{2}. While we clearly still have f, it is not obvious how we should get x. This is where the idea of midpoints comes in again.

To estimate the mean and standard deviation from a histogram, first turn the histogram into a table, then add a column of the midpoints of each class labelled x. Then, create columns fx and fx^{2} and find the totals of all of the columns. Finally, use these totals in the formulas for mean and standard deviation.

Recall: The formulas:

\text{mean}=\dfrac{\sum{fx}}{\sum{f}}

\text{variance}=\dfrac{\sum{fx^{2}}}{\sum{f}}-\text{mean}^{2}

\text{standard deviation}=\sqrt{\text{variance}}

Estimating the Median from a Histogram

To estimate the median from a histogram we use linear interpolation. This is where we assume that within each block, the frequency is evenly spaced.

To find the median, first find \sum{f} and divide it by 2 to find the position of the median (since this is an estimate, if we obtain a decimal we can treat it as if it is a whole number position). Then, find which block the position falls into. Then, within that block, find where it lies.

For example, if the median is the 7th position of a block with 10 values of length 5, then you would add \dfrac{7\times 5}{10}=3.5 to the lower bound of the block to find the median.