In a frequency table, f counts how many times a value x occurs. Thus x= fx f: weight each value by its frequency. For grouped intervals, use the midpoint as a representative value; the result is an estimate because the exact observations are lost. Example **Time t (min)** | **Frequency** | **Midpoint** 0<t 10 | 4 | 5 10<t 20 | 8 | 15 20<t 30 | 8 | 25 Estimated mean= 4(5)+8(15)+8(25)20=17 min. Equal class widths allow frequency to be the bar height. Bars touch because the intervals are adjacent; the horizontal axis shows class boundaries. Unlike a bar chart, the intervals are quantitative, not separate categories. The boundary 10 belongs to the first interval here, not both. There are 12 observations at or below 20 minutes. The last two classes tie for highest frequency. A grouped table does not reveal the exact smallest or largest value.

In a frequency table, \(f\) counts how many times a value \(x\) occurs. Thus \(\overline{x}=\dfrac{\sum fx}{\sum f}\): weight each value by its frequency. For grouped intervals, use the midpoint as a representative value; the result is an estimate because the exact observations are lost.

\(\text{Estimated mean}=\dfrac{4(5)+8(15)+8(25)}{20}=17\text{ min}\).

Equal class widths allow frequency to be the bar height. Bars touch because the intervals are adjacent; the horizontal axis shows class boundaries. Unlike a bar chart, the intervals are quantitative, not separate categories.

The boundary \(10\) belongs to the first interval here, not both. There are \(12\) observations at or below \(20\) minutes. The last two classes tie for highest frequency. A grouped table does not reveal the exact smallest or largest value.

Time \(t\) (min)	Frequency	Midpoint
\(0<t\le10\)	\(4\)	\(5\)
\(10<t\le20\)	\(8\)	\(15\)
\(20<t\le30\)	\(8\)	\(25\)

Frequency tables and equal-interval histograms

Frequency tables and equal-interval histograms