02-02: Exercises — Graphical Displays

Notes reference: 02-02: Graphical Displays

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Q1: Pick the right graph

Name the best graph for each task.

1. Show what share of total revenue each of 4 product lines contributes.
2. Compare the distribution of salaries in three departments.
3. Show the shape of 200 reaction times.
4. Show how many customers are at or below each spending level.
5. Show 15 test scores while keeping the original values visible.
6. Show whether study hours and exam score move together.
7. Show monthly website visits over three years.
8. Rank 8 defect causes and show the cumulative share.

Solution

1. PIE CHART (or a sorted bar chart — easier to compare 4 slices)
2. SIDE-BY-SIDE BOXPLOTS
3. HISTOGRAM
4. OGIVE (cumulative frequency graph)
5. STEM-AND-LEAF PLOT — shape plus every original value
6. SCATTERPLOT
7. TIME SERIES PLOT (line chart)
8. PARETO CHART

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Q2: Bar chart or histogram?

For each, say which applies and whether the bars touch.

Data	Graph	Bars touch?
Blood type of 200 donors	?	?
Weight of 200 donors, grouped into classes	?	?
Number of children per family (0,1,2,3,4)	?	?

Solution

Data	Graph	Bars touch?
Blood type	Bar chart — nominal categories	No — separated
Weight, grouped	Histogram — continuous scale	Yes — touching
Number of children	Bar chart — discrete counts, treated as categories	No (though some texts allow a histogram-style plot)

' To make histogram bars touch in Excel:
'   right-click the series ▸ Format Data Series ▸ Gap Width = 0%

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Q3: Build a stem-and-leaf plot

Twenty-two scores:

34  38  41  43  45  47  48  50  52  53  55
56  57  58  60  62  63  65  67  71  74  78

Draw the stem-and-leaf plot with a key, then state the shape.

Solution

Stem | Leaf
-----+---------------------------
   3 | 4 8
   4 | 1 3 5 7 8
   5 | 0 2 3 5 6 7 8
   6 | 0 2 3 5 7
   7 | 1 4 8

Key: 5 | 2  =  52

Shape: roughly SYMMETRIC and unimodal, peaking in the 50s.
       n = 2 + 5 + 7 + 5 + 3 = 22  ✓

stem(c(34,38,41,43,45,47,48,50,52,53,55,
       56,57,58,60,62,63,65,67,71,74,78))

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Q4: Read a boxplot

A boxplot of house prices (in $000s) shows: minimum whisker 180, Q1 245, median 290, Q3 380, upper whisker 520, with two dots at 720 and 850.

1. What is the IQR?
2. What are the outlier fences?
3. Are the two dots correctly flagged?
4. Describe the skew.

Solution

1. IQR = Q3 − Q1 = 380 − 245 = 135

2. Lower fence = 245 − 1.5(135) = 245 − 202.5 = 42.5
   Upper fence = 380 + 1.5(135) = 380 + 202.5 = 582.5

3. 720 and 850 both exceed 582.5  →  YES, correctly flagged as outliers.
   The upper whisker stops at 520, the largest value INSIDE the fence. ✓

4. Median (290) sits LEFT of the box centre ((245+380)/2 = 312.5),
   the right whisker (380→520 = 140) is longer than the left (245→180 = 65),
   and both outliers are on the high side.
   →  RIGHT-SKEWED (positively skewed) — the usual shape for house prices.
   →  Report the MEDIAN and IQR, not the mean and SD.

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Q5: Frequency polygon and ogive

Using the commute-time distribution from 02-01:

Class	Xm	f	Upper boundary	Cum f
10–19	14.5	3	19.5	3
20–29	24.5	4	29.5	7
30–39	34.5	8	39.5	15
40–49	44.5	7	49.5	22
50–59	54.5	5	59.5	27
60–69	64.5	3	69.5	30

State exactly which (x, y) pairs you plot for (a) the frequency polygon and (b) the ogive.

Solution

(a) FREQUENCY POLYGON — plot (MIDPOINT, frequency), closing on the axis:

    (4.5, 0)  (14.5, 3)  (24.5, 4)  (34.5, 8)  (44.5, 7)  (54.5, 5)  (64.5, 3)  (74.5, 0)
     ↑ one class below                                                  ↑ one class above
     with f = 0                                                         with f = 0

(b) OGIVE — plot (UPPER BOUNDARY, cumulative frequency), starting at 0:

    (9.5, 0)  (19.5, 3)  (29.5, 7)  (39.5, 15)  (49.5, 22)  (59.5, 27)  (69.5, 30)
     ↑ the LOWER boundary of the first class, at cumulative frequency 0

The two most common errors: plotting the polygon at boundaries instead of midpoints, and plotting the ogive at limits instead of boundaries.

' Both are Scatter with Straight Lines and Markers.
' Build two columns (x and y) exactly as listed above, then Insert ▸ Scatter.

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Q6: Diagnose the shape

Match each description to its shape, and say which centre and spread to report.

1. Household income for a city
2. Heights of adult women
3. Scores on a very easy test where most students scored 90+
4. The last digit of 500 phone numbers
5. Heights of a mixed group of adult men and women

Solution

#	Shape	Report
1	Right-skewed — a long tail of high earners	Median, IQR
2	Symmetric / bell-shaped	Mean, standard deviation
3	Left-skewed — a tail of low scores	Median, IQR
4	Uniform — each digit equally likely	Range; a mean is meaningless here
5	Bimodal — two overlapping subgroups	Split by sex and describe each separately

Rule of thumb that covers 1–3:  the MEAN gets dragged toward the long tail.
  mean > median  →  right-skewed
  mean < median  →  left-skewed

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Q7: Spot the misleading graph

A press release shows a bar chart of quarterly sales: Q1 = 4.8 M, Q2 = 5.0 M, Q3 = 5.1 M. The vertical axis runs from 4.7 to 5.2 and Q3's bar looks four times taller than Q1's. The headline reads "Sales quadruple!"

What is wrong, and how would you redraw it?

Solution

PROBLEM   TRUNCATED AXIS. A bar chart encodes value as BAR LENGTH, so its
          axis must start at 0. Starting at 4.7 turns a 6% rise (4.8 → 5.1)
          into a bar that is visually 4× longer.

The real change:  (5.1 − 4.8) / 4.8 = 6.25% growth over two quarters.

FIX  1.  Start the vertical axis at 0.
     2.  If the differences are then too small to see, use a LINE CHART
         (which encodes value by position, where a truncated axis is
         conventional and acceptable) and label the axis clearly.
     3.  Or plot the percentage change directly, which is what the
         headline is actually about.
     4.  Rewrite the headline: "Sales up 6% since Q1".

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Q8: Reproduce a figure in all three tools

Produce a histogram of x with 6 classes plus a boxplot beneath it, in Excel, R, and Python.

Solution

' HISTOGRAM
'   Option A: Insert ▸ Insert Statistic Chart ▸ Histogram,
'             then right-click the horizontal axis ▸ Format Axis ▸
'             Number of bins = 6
'   Option B: Data ▸ Data Analysis ▸ Histogram with a Bin Range of
'             upper class limits, tick Chart Output, then set
'             Format Data Series ▸ Gap Width = 0
'
' BOXPLOT
'   Select the data ▸ Insert ▸ Insert Statistic Chart ▸ Box and Whisker
'   Format Data Series ▸ tick "Show mean markers"

par(mfrow = c(2, 1), mar = c(4, 4, 2, 1))
hist(x, breaks = 6, col = "#8A5FBF", border = "white",
     main = "Distribution of x", xlab = "")
boxplot(x, horizontal = TRUE, col = "#0FA3A3", xlab = "x")
par(mfrow = c(1, 1))

# ggplot2 version
library(ggplot2)
ggplot(data.frame(x), aes(x)) +
  geom_histogram(bins = 6, fill = "#5B2A86", colour = "white") +
  theme_minimal()

import matplotlib.pyplot as plt
fig, (a1, a2) = plt.subplots(2, 1, figsize=(7, 5), sharex=True,
                             gridspec_kw={"height_ratios": [3, 1]})
a1.hist(x, bins=6, color="#8A5FBF", edgecolor="white")
a1.set(ylabel="Frequency", title="Distribution of x")
a2.boxplot(x, vert=False, showmeans=True)
a2.set(xlabel="x", yticks=[])
plt.tight_layout(); plt.show()

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