Lecture 2 · Week 2 · Lind, Marchal & Wathen ch. 2 · about 45 min

Frequency Tables and Graphs

How to summarize data with tables and graphs: frequency and relative frequency tables for qualitative data, bar and pie charts, frequency distributions built with the 2 to the k rule, histograms, frequency polygons, and cumulative distributions.

By the end you can

Build frequency and relative frequency tables for qualitative data
Read and choose between bar charts and pie charts
Construct a frequency distribution: number of classes, class interval, class limits and midpoints
Draw and interpret histograms and frequency polygons
Use cumulative frequency polygons to estimate values and percentages
Avoid the classic exam traps in frequency tables and graphs

Summarizing qualitative data with frequency tables

Descriptive statistics organize data so that the general pattern, the values where data concentrate, and extreme values all become visible. The first tool is the frequency table.

Qualitative variables are nonnumeric labels: dealership location, vehicle type, method of payment. To summarize them, we count how many observations carry each label.

Example (the chapter’s LO2-1). The Applewood Auto Group sold 180 vehicles last month at four dealerships. Counting sales by location gives the frequency table:

Location	Frequency (cars)	Relative frequency
Kane	52	$52/180 = 0.289$
Olean	40	$40/180 = 0.222$
Sheffield	45	$45/180 = 0.250$
Tionesta	43	$43/180 = 0.239$
Total	180	1.000

A relative frequency is a class frequency divided by the total number of observations, $f/n$ . It answers “what fraction of the whole is this class?” Multiply by 100 for the percentage: Kane’s share is 28.9% of all sales.

Relative frequencyFormula sheet →

\text{relative frequency} = \frac{f}{n}

$f$: class frequency
$n$: total number of observations

Use it when you want the share of the total that falls in a class rather than the raw count.

Bar charts and pie charts

The frequency table is the summary; a chart is the picture. For qualitative data there are two standard pictures.

The chapter’s example is a SkiLodges.com usability survey: 200 randomly selected users rated the ease of navigation of a new website as awesome (102), excellent (58), good (30), or poor (10). The rating scale is ordinal — the categories are ranked — but the distance between neighboring ratings is unknown, so we cannot say “good” is twice “poor”.

Rating	Frequency	Relative frequency	Percent
Awesome	102	0.51	51%
Excellent	58	0.29	29%
Good	30	0.15	15%
Poor	10	0.05	5%
Total	200	1.00	100%

Because a pie slice’s share of the circle is its relative frequency, the central angle of a slice is the relative frequency times 360 degrees:

Pie slice angleFormula sheet →

\text{slice angle} = \text{relative frequency} \times 360^\circ

$relative frequency$: class share of the total, as a fraction
$360^\circ$: the full circle

Use it when you are drawing a pie chart and need the central angle of one class's slice.

Example: Applewood’s 180 vehicle sales by type are sedan 72 (40%), SUV 54 (30%), compact 27 (15%), truck 18 (10%), hybrid 9 (5%). The sedan slice is $0.40 \times 360^\circ = 144^\circ$ , and the hybrid slice is only $0.05 \times 360^\circ = 18^\circ$ .

Frequency distributions for quantitative data

When the variable is quantitative (profit, age, lead time), the summary is a frequency distribution. The mechanics are similar to a frequency table, but now the classes are intervals of numbers, and building them takes four steps.

The chapter’s running example: Applewood Auto Group earned a profit on each of the 180 vehicles sold. The profits are raw data; scanning the list we can find the minimum (294) and the maximum (3,292) dollars, but patterns are hard to see. The goals of the summary: the typical profit, the extremes, and where profits cluster.

Step 1 — decide the number of classes: the 2 to the k rule

The 2 to the k ruleFormula sheet →

2^k \gt n

$k$: number of classes
$n$: number of observations

Use it when you must choose how many classes a frequency distribution should have; pick the smallest k that satisfies the rule.

For Applewood, $n = 180$ . Try $k = 7$ : $2^7 = 128$ , which is less than 180, so 7 classes are too few. Try $k = 8$ : $2^8 = 256$ , which is greater than 180. So we use 8 classes.

Step 2 — determine the class interval

Class intervalFormula sheet →

i = \frac{\mathrm{Maximum} - \mathrm{Minimum}}{k}

$i$: class interval
$Maximum$: largest data value
$Minimum$: smallest data value
$k$: number of classes

Use it when the number of classes is chosen and you need the width of each class.

For Applewood: $i = (3292 - 294)/8 = 374.75$ . In practice, round the interval up to a convenient number, usually a multiple of 10 or 100 — here 400. Rounding up matters: rounding down would leave part of the data range uncovered.

Step 3 — set the class limits

Class limits must not overlap, or an observation could fall into two classes. The book’s convention is “up to”: in the classes “1,300 up to 1,400” and “1,400 up to 1,500”, the value 1,400 clearly belongs to the second class.

Because the rounded interval covers a bit more range than needed, the excess is split between the two tails. A useful guide is to start the first class at a convenient lower limit, ideally a multiple of the interval. For Applewood, eight classes of width 400 cover $8 \times 400 = 3{,}200$ while the data range is only $3{,}292 - 294 = 2{,}998$ ; starting at 200 and ending at 3,400 splits the extra 202 dollars between the tails.

Step 4 — tally, and define the midpoint

Tally each observation into its class and count. The number of observations in a class is its class frequency; frequencies must sum to the total number of observations. Each class also gets a class midpoint, the single value that best represents the class.

Class midpointFormula sheet →

\mathrm{midpoint} = \frac{L + U}{2}

$L$: lower limit of the class
$U$: upper limit of the same class

Use it when you need one typical value to represent a class, for instance to plot a frequency polygon.

Example: the first class runs from 200 up to 600, so its midpoint is $(200 + 600)/2 = 400$ . The class 1,800 up to 2,200 has midpoint $(1{,}800 + 2{,}200)/2 = 2{,}000$ .

Profit (dollars)	Midpoint	Frequency
200 up to 600	400	8
600 up to 1,000	800	11
1,000 up to 1,400	1,200	23
1,400 up to 1,800	1,600	38
1,800 up to 2,200	2,000	45
2,200 up to 2,600	2,400	32
2,600 up to 3,000	2,800	19
3,000 up to 3,400	3,200	4
Total		180

Relative frequency distributions

A frequency distribution counts; a relative frequency distribution divides. Each class frequency is divided by $n$ , the total number of observations, and the relative frequencies sum to 1 (100%). This is what lets you read statements like “12.8% of the vehicles earned a profit between 1,000 and 1,400.”

Cumulative relative frequencyFormula sheet →

\text{cumulative relative frequency} = \frac{\text{cumulative frequency}}{n}

$cumulative frequency$: number of observations below a class upper limit
$n$: total number of observations

Use it when you want the share of observations below a value rather than a single class share.

For Applewood, dividing each frequency by 180 gives:

Profit (dollars)	Frequency	Relative frequency	Found by
200 up to 600	8	0.044	$8/180$
600 up to 1,000	11	0.061	$11/180$
1,000 up to 1,400	23	0.128	$23/180$
1,400 up to 1,800	38	0.211	$38/180$
1,800 up to 2,200	45	0.250	$45/180$
2,200 up to 2,600	32	0.178	$32/180$
2,600 up to 3,000	19	0.106	$19/180$
3,000 up to 3,400	4	0.022	$4/180$
Total	180	1.000

Reading the table: one quarter of the vehicles (0.250) earned between 1,800 and 2,200 — the most common class; about 87% earned between 1,000 and 3,000. A relative frequency histogram would have exactly the same shape as the histogram of counts, only the vertical axis would show percentages instead of vehicles.

The same idea applies to any distribution. Consider the chapter’s electronics-imports example, where 50 suppliers report annual imports:

Imports (millions of dollars)	Number of suppliers	Relative frequency
2 up to 5	6	0.12
5 up to 8	13	0.26
8 up to 11	20	0.40
11 up to 14	10	0.20
14 up to 17	1	0.02
Total	50	1.00

Histograms

Reading the histogram gives the same statements as reading the table: profits range from about 200 to 3,400 dollars; the class interval is 400; profits concentrate between 1,000 and 3,000 (157 of 180 vehicles, about 87%); the tallest bar is the 1,800 up to 2,200 class with 45 vehicles and midpoint 2,000 — the typical profit of the most common class. If you plot relative frequencies instead of counts, the shape is identical and only the vertical axis changes to percentages.

Describing the shape of a distribution

Every histogram has a shape, and naming it is a standard exam question. The three benchmark shapes:

Symmetric: the left and right sides roughly mirror each other.
Right-skewed (positively skewed): a long tail stretches to the right — a few unusually large values.
Left-skewed (negatively skewed): a long tail stretches to the left — a few unusually small values.

Frequency polygons

Construction with the Applewood profits: compute each class midpoint, plot the points (400, 8), (800, 11), (1,200, 23), (1,600, 38), (2,000, 45), (2,400, 32), (2,800, 19), (3,200, 4), and connect them. To close the polygon on the axis, anchor at zero frequency with two extra midpoints: one class interval below the lowest midpoint ( $400 - 400 = 0$ ) and one above the highest ( $3{,}200 + 400 = 3{,}600$ ), giving endpoints (0, 0) and (3,600, 0).

Cumulative frequency distributions

A frequency distribution answers “how many observations are in this class?” A cumulative frequency distribution answers a different, very practical question: “how many observations are below this value?” — for example, how many vehicles earned a profit of less than 1,400 dollars.

Profit (dollars)	Frequency	Cumulative frequency	Cumulative relative frequency
Less than 600	8	8	$8/180 = 0.044 = 4.4\%$
Less than 1,000	11	19	$19/180 = 0.106 = 10.6\%$
Less than 1,400	23	42	$42/180 = 0.233 = 23.3\%$
Less than 1,800	38	80	$80/180 = 0.444 = 44.4\%$
Less than 2,200	45	125	$125/180 = 0.694 = 69.4\%$
Less than 2,600	32	157	$157/180 = 0.872 = 87.2\%$
Less than 3,000	19	176	$176/180 = 0.978 = 97.8\%$
Less than 3,400	4	180	$180/180 = 1.000 = 100\%$

The running totals: 8 vehicles earned less than 600; those 8 plus the 11 in the next class make 19 below 1,000; plus 23 makes 42 below 1,400; and so on until all 180 are below 3,400. Read across the fourth column: 44.4% of the vehicles earned less than 1,800 dollars.

To graph it, plot the upper limit of each class against the cumulative frequency, beginning with a point at zero: (200, 0), (600, 8), (1,000, 19), (1,400, 42), (1,800, 80), (2,200, 125), (2,600, 157), (3,000, 176), (3,400, 180). Connect the points to get the cumulative frequency polygon. The line always rises from left to right (nothing is below the minimum), and the right-hand axis can carry the percentage scale instead of the count scale.

Two more readings from the same graph: 60 vehicles correspond to about 1,600 dollars, and a profit level of 2,000 dollars corresponds to roughly $80 + 0.5 \times 45 = 102.5$ vehicles, that is about 57% of the total — close to the chapter’s “about 56%”.

Choosing the right table or graph

Question to answer	Variable	Tool
Summarize the classes and their counts	qualitative	frequency table
Compare shares of a whole	qualitative	pie chart
Compare counts across classes	qualitative	bar chart
Summarize numeric data in intervals	quantitative	frequency distribution
See shape, highs and lows at a glance	quantitative	histogram
Compare two distributions	quantitative	frequency polygon
“How many below a value?”	quantitative	cumulative frequency polygon

Exam trap · Exam traps to remember

Gap or no gap? Bar charts (qualitative) leave gaps between bars; histograms (quantitative) have adjacent bars.
“Up to” boundaries: the upper limit starts the next class. 1,400 belongs to “1,400 up to 1,500”, not to “1,300 up to 1,400”.
Round the interval up to a convenient multiple such as 10 or 100. Never round down — part of the range would be left uncovered.
Relative frequencies sum to 1, and the final cumulative relative frequency equals 1 (100%). If it does not, a class was dropped, double-counted, or divided wrongly.
Sample variance divides by $n - 1$ , not by $n$ — the classic first-exam trap once measures of variation appear; the population variance divides by $N$ .
Skewed data: the mean follows the tail, the median does not. Right-skewed: mean above median. Left-skewed: mean below median.

Exam practice

Summary and review

A frequency table summarizes a qualitative variable: mutually exclusive, collectively exhaustive classes, with counts that sum to the total.
A relative frequency is $f/n$ ; shares are comparable across groups and always sum to 1 (100%).
Bar charts show counts for qualitative classes with gaps between bars; pie charts show shares, with slice angle $= \text{relative frequency} \times 360^\circ$ .
A frequency distribution summarizes a quantitative variable. Build it in four steps: (1) number of classes by the $2^k$ rule, (2) class interval $(\mathrm{Max} - \mathrm{Min})/k$ rounded up, (3) non-overlapping class limits in “up to” form, (4) tally into classes.
The class midpoint, $(L + U)/2$ , is the typical value of a class; frequencies across classes sum to n (the total number of observations).
A relative frequency distribution divides each class frequency by n; it has the same shape as the count distribution.
A histogram plots class frequencies as adjacent bars over a continuous axis; it reveals the range, the concentration, and the shape: symmetric, right-skewed, or left-skewed.
Skew moves the mean: right-skewed means mean greater than median; left-skewed means mean less than median. The median resists extremes.
A frequency polygon joins (midpoint, frequency) points, anchored at zero frequency one interval beyond each end; it is the tool for overlaying and comparing distributions.
A cumulative frequency polygon plots (upper limit, cumulative frequency) and answers “less than” questions, for example: 75% of vehicles earned about 2,300 dollars or less.
Notation: population $N, \mu, \sigma$ ; sample $n, \bar{x}, s$ . Sample variance divides by $n - 1$ , population variance by $N$ .