Grouped frequency tables
When there is a lot of data, we group values into class intervals such as $10\text{–}19$, $20\text{–}29$. Two useful quantities for each class are the midpoint (the average of the two class limits) and the class boundaries (halfway between adjacent classes). The modal class is the class with the highest frequency.
Key formula
$\text{midpoint}=\frac{\text{lower limit}+\text{upper limit}}{2},\qquad \text{class width}=\text{upper boundary}-\text{lower boundary}$
Histograms
A histogram displays grouped data using bars whose heights represent frequency. Unlike a bar chart the bars touch, because the horizontal axis is a continuous number scale drawn using the class boundaries.
Cumulative frequency and the ogive
Cumulative frequency is the running total of frequencies up to the upper boundary of each class. When we plot cumulative frequency against the upper boundary and join the points with a smooth curve, we obtain the ogive (cumulative frequency graph), from which the median and quartiles can be read.
Example
For the class $20\text{–}29$: the midpoint is $\dfrac{20+29}{2}=24.5$, the upper boundary is $29.5$ and the lower boundary is $19.5$, so the class width is $29.5-19.5=10$.
Remember
Boundaries, not limits, are used for widths and for the horizontal scale of a histogram.