B. Probability Distributions · Continuous Distributions and the Normal Distribution

Frequency Tables, Histograms and Frequency Polygons

B2.4About 70 minutes

Learning goals

I can organize continuous data into a frequency table with equal class intervals.
I can draw a histogram and a frequency polygon, including relative frequencies.
I can explain how the choice of interval width changes the shape a histogram or polygon shows.
I can choose a sensible number of classes for a data set.

Words to know

class width
The length of each class interval, for example 5 cm for the class 25 to 30 cm.

frequency table
A table listing each class interval with the number of data values (frequency) in it.

relative frequency
The fraction (or percent) of all data values in a class: frequency ÷ total.

histogram
A graph of a frequency distribution for continuous data: touching bars over the class intervals, with heights equal to the frequencies (or relative frequencies).

frequency polygon
A broken-line graph joining the points (class midpoint, frequency), closed off by adding an empty class at each end so the graph meets the horizontal axis.

Forty maple seedlings

A tree nursery measures 40 sugar maple seedlings, all 16 months old, to see how tall they have grown. A list of 40 numbers is hard to read. In this lesson you will turn it into a frequency table, a histogram and a frequency polygon, and you will see how the width of the classes changes the picture.

Remember from Grade 9: in the Grade 9 data unit you drew histograms and described shapes (symmetric, skewed, uniform). Here you will build them carefully for continuous data and add the frequency polygon, which leads straight to the smooth curves in the next lesson.

Seedlings 1 to 10	11 to 20	21 to 30	31 to 40
16.3	27.3	33.7	36.6
16.8	27.8	34.2	36.7
17.0	28.0	34.2	37.0
23.4	28.9	34.7	37.7
23.6	29.2	34.8	39.7
24.0	29.7	35.5	41.3
25.3	30.0	35.6	42.0
26.0	30.1	35.8	42.8
26.2	31.0	36.5	43.0
26.4	31.6	36.6	47.5

Heights (cm) of 40 maple seedlings, sorted from shortest to tallest (read down each column). Smallest 16.3 cm, largest 47.5 cm.

Step 1: Build a frequency table

Find the range. 47.5 − 16.3 = 31.2 cm.
Choose the number of classes. For 30 to 100 values, about 5 to 10 classes usually works. Too few hides the shape; too many makes it jagged.
Choose a convenient width. 31.2 ÷ 7 ≈ 4.5, so round up to a friendly 5 cm.
Set the boundaries. Start just below the smallest value at a round number: 15, 20, 25, ..., 50. Use the rule lower boundary ≤ x < upper boundary, so a value of exactly 30.0 goes in 30 to 35, never in two classes.
Tally each value and total the frequencies. They must add to 40.

Height (cm)	Midpoint	Frequency	Relative frequency
15 to 20	17.5	3	0.075
20 to 25	22.5	3	0.075
25 to 30	27.5	10	0.250
30 to 35	32.5	9	0.225
35 to 40	37.5	10	0.250
40 to 45	42.5	4	0.100
45 to 50	47.5	1	0.025
Total		40	1.000

Frequency table with class width 5 cm. Relative frequency = frequency ÷ 40; the relative frequencies add to 1.

Step 2: Histogram and frequency polygon

Histogram: class boundaries on the horizontal axis, frequency on the vertical axis, bars that touch.
Frequency polygon: plot a point at the midpoint of each class at the height of its frequency, and join the points with straight segments. Then add an empty class at each end (midpoints 12.5 and 52.5, frequency 0) so the polygon comes down to the axis.

A neat fact: when the classes have equal width, the area under the frequency polygon equals the total area of the histogram bars. Each small triangle the polygon cuts off a bar is matched by an equal triangle it adds beside the bar.

If you plot relative frequency instead of frequency, the graph has exactly the same shape; only the vertical scale changes.

Histogram (bars) and frequency polygon (line) for the 40 seedlings, class width 5 cm. The polygon joins the class midpoints and is closed at (12.5, 0) and (52.5, 0).

Step 3: How the interval width changes the picture

The same 40 heights can look quite different depending on the class width.

Width 10 cm (4 classes): smooth, but too coarse. You only see "most seedlings are 20 to 40 cm"; the details are hidden.
Width 5 cm (7 classes): a clear single peak around 25 to 40 cm and a short tail to the right. This is a good choice for 40 values.
Width 2 cm (16 classes): very jagged, with empty classes and spikes. With only 40 values, many classes hold 0, 1 or 2 seedlings, so random bumps dominate.

With more data, narrower classes become useful: for 4000 seedlings, a width of 2 cm would show a smooth shape. This is the key idea of the next lesson: as the number of data values grows and the class width shrinks, the frequency polygon gets closer and closer to a smooth curve.

The same 40 seedlings with class widths 10, 5 and 2 cm. Wide classes hide detail; very narrow classes make the polygon jagged. Width 5 cm shows the shape best for this small data set.

Worked example

Example 1: Reading the frequency table

Use the width-5 table for the seedlings. (a) What percent of the seedlings are under 30 cm tall? (b) What percent are at least 35 cm tall? (c) Which class is the modal class?

(a) Under 30 cm: 3 + 3 + 10 = 16 seedlings. 16 ÷ 40 = 0.40.
(b) At least 35 cm: 10 + 4 + 1 = 15 seedlings. 15 ÷ 40 = 0.375.
(c) Two classes tie with the highest frequency, 10: 25 to 30 and 35 to 40.

(a) 40% (b) 37.5% (c) The classes 25 to 30 cm and 35 to 40 cm tie (frequency 10 each). A tie like this, with a dip between, is often just random variation in a small sample.

Worked example

Example 2: Points for the frequency polygon

List the points you would plot for the frequency polygon of the seedling data with width 10 cm (classes 10 to 20, 20 to 30, 30 to 40, 40 to 50 with frequencies 3, 13, 19, 5).

Midpoints of the classes: 15, 25, 35, 45.
Points: (15, 3), (25, 13), (35, 19), (45, 5).
Add an empty class at each end: midpoints 5 and 55 with frequency 0.

(5, 0), (15, 3), (25, 13), (35, 19), (45, 5), (55, 0).

Worked example

Example 3: Area of a relative frequency histogram

Find the total area of the bars in a histogram of the seedling data, width 5 cm, if the bar heights are the relative frequencies divided by the class width (relative frequency per cm).

Each bar's height is (relative frequency) ÷ 5 and its width is 5.
Area of one bar = width × height = 5 × (relative frequency ÷ 5) = relative frequency.
So the total area = sum of the relative frequencies = 1.

The total area is 1. When the vertical axis shows relative frequency per unit, area equals proportion. This is exactly how a continuous probability distribution works, as you will see in the next lesson.

Just for fun

A histogram and a bar graph go to a dance. The bar graph keeps a polite space between every partner. The histogram says, "Sorry, I'm continuous," and holds everyone's hand.

Hands-on activity

Measure 30 of something

You need: a ruler or measuring tape, a pencil, paper (or a free spreadsheet) and 30 similar objects you can measure, for example 30 leaves from the same tree (in fall), 30 baby carrots, 30 crayons or pencils, or the hand spans of family and friends.

Measure each object to the nearest millimetre and record the 30 values.
Find the range and choose a class width that gives about 5 to 8 classes.
Make a frequency table with midpoints and relative frequencies.
Draw the histogram and the frequency polygon on the same axes.
Redraw the histogram with a width twice as large and one half as large.

What to look for: the middle width should show the clearest shape. The wide one looks blocky; the narrow one looks spiky with gaps. Describe the shape you see (one peak? symmetric? a long tail?).

Your turn

A data set has smallest value 12.4 and largest value 61.0. You want about 6 classes. Suggest a class width and the first two classes.

Using the rule lower ≤ x < upper, which class (25 to 30 or 30 to 35) does a value of 30.0 belong to?

Snowfall depths (cm) for 20 days: 0 to 5 (6), 5 to 10 (7), 10 to 15 (4), 15 to 20 (2), 20 to 25 (1). Find the relative frequency of each class.

For the snowfall table, list all the points of the frequency polygon.

Describe the shape of the snowfall distribution.

Why does a histogram with 16 classes look jagged when there are only 40 data values?

More practice: warm-up

What is plotted on the horizontal axis of a frequency polygon?

What is the relative frequency of a class with 12 values out of 48?

What must the frequencies in a table add to?

Do the bars in a histogram of continuous data touch?

A class width of 5 starts at 20. What are the next two classes?

More practice: on your own

Times (s) for 30 runners: 50 to 55 (2), 55 to 60 (6), 60 to 65 (11), 65 to 70 (8), 70 to 75 (3). What percent ran under 60 s? What percent took 65 s or more?

For the runner data, name the modal class and describe the shape.

The runner data are regrouped with width 10 s (50 to 60, 60 to 70, 70 to 80). Find the new frequencies. What detail is lost?

Explain why the area under a frequency polygon equals the area of the histogram (equal class widths).

A histogram's vertical axis shows relative frequency per cm, and the classes are 5 cm wide. A bar has height 0.05. What proportion of the data is in that class?

Challenge

Two classmates make histograms of the same 200 heights. One uses width 1 cm and sees many spikes; the other uses width 20 cm and sees two bars. Advise them, using the range of 45 cm.

A histogram uses unequal classes: 0 to 10 (frequency 20) and 10 to 40 (frequency 30). Why would plotting frequency as the bar height mislead the reader? What should the heights be instead?

Using the seedling frequency table, estimate the median height. Assume the values in the median class are spread evenly.

Check your understanding

Quiz

A frequency polygon joins which points?

A data set has range 36 and you want about 6 classes. Which class width is most sensible?

A class contains 9 of 60 data values. What is its relative frequency?

What usually happens to a histogram of 40 values when the class width is made very small?

Why is an empty class added at each end of a frequency polygon?

Using the rule lower ≤ x < upper, which class does the value 40.0 belong to?

For parents and teachers

Big idea (B2.4): continuous data are organized into equal class intervals. A histogram (touching bars) and a frequency polygon (a line through class midpoints) show the shape, and the class width strongly affects what you see.

How to help: Do the activity together: measuring 30 leaves, carrots or crayons takes 15 minutes and makes the ideas concrete. Ask your teen why they chose their class width, and what changes when they double or halve it. A spreadsheet's histogram tool is a good check.

Common difficulties:

Classes that overlap (20 to 25 and 25 to 30 both "containing" 25). Use lower ≤ x < upper.
Plotting polygon points at class boundaries instead of midpoints, or forgetting the empty end classes.
Choosing far too many or too few classes.
Leaving gaps between histogram bars.