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 $2times 8=16$ 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 $7$th position of a block with $10$ values of length $5$, then you would add $dfrac{7times 5}{10}=3.5$ to the lower bound of the block to find the median.