Construct and interpret histograms

Detailed Explanation: Construct and interpret histograms

A histogram shows how numerical data are distributed. It groups values into contiguous intervals, called bins. Each bin is represented by a bar, and the bar’s height shows the frequency—the number of data values in that interval. The bars touch because the intervals are next to one another.

Example: A school records the waiting times, in minutes, for 24 students at a bus stop.

Step 1: Check the intervals

The intervals have equal width:

They do not overlap, and together they cover the possible waiting times. The wording “less than” prevents a value from belonging to two intervals. For example, a waiting time of $10$ minutes belongs in the interval $10$ to less than $15$.

Step 2: Set up the axes

• Put the waiting-time intervals on the horizontal axis.
• Put frequency on the vertical axis.
• Choose a scale that reaches at least $8$, such as $0, 2, 4, 6, 8$.

Step 3: Draw the bars

Draw one bar over each interval with height equal to its frequency:

• $0$ to less than $5$: height $3$
• $5$ to less than $10$: height $6$
• $10$ to less than $15$: height $8$
• $15$ to less than $20$: height $4$
• $20$ to less than $25$: height $2$
• $25$ to less than $30$: height $0$
• $30$ to less than $35$: height $1$

The bars should touch, including the empty interval from $25$ to less than $30$. The frequencies add to

which matches the number of students.

Step 4: Interpret the histogram

• The modal class, or interval with the greatest frequency, is $10$ to less than $15$ minutes.
• The waiting times are mostly between $5$ and $20$ minutes, so the center is roughly around $10$ to $15$ minutes.
• The data spread from about $0$ to $35$ minutes.
• There is a gap from $25$ to less than $30$ minutes because that interval has frequency $0$.
• The isolated value in the $30$ to less than $35$ interval may be a possible high outlier. We would check the original data before deciding whether it truly is an outlier.
• The distribution has a longer tail toward larger waiting times, so it is slightly skewed right.

A histogram is different from a bar graph because its horizontal intervals are numerical and ordered, and its bars touch. Bar graphs usually have separate categories, so their bars have spaces between them.