Graphical Representations of Data, Mean, Median and

Standard Deviation

In this class we will consider graphical representations of the

distribution of a set of data. The goal is to identify the range of

values and the most likely values as well as properties like

symmetry, uni- or multi-modality, tail behavior, etc.

To quantify the central value of the distribution of a given sample

we define the average and the median. To quantify the spread

(dispersion) of the sample with respect to its central value we

define the standard deviation.

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Pie Charts

The pie

charts corre-

spond to the

proportion

of ice-cream

flavors sold

annually

by a given

brand

Blueberry

Cherry

Apple

Boston CreamOther

Vanilla Cream

Blueberry

Cherry

Apple

Boston CreamOther

Vanilla Cream

Blueberry

Cherry

Apple

Boston CreamOther

Vanilla Cream

Blueberry

Cherry

Apple

Boston CreamOther

Vanilla Cream

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Pie Charts are a bad idea!

From the R manual page for the pie function:

Pie charts are a very bad way of displaying information.

The eye is good at judging linear measures and bad at

judging relative areas. A bar chart or dot chart is a

preferable way of displaying this type of data.

Cleveland (1985), page 264: "Data that can be shown by

pie charts always can be shown by a dot chart. This means

that judgements of position along a common scale can be

made instead of the less accurate angle judgements." This

statement is based on the empirical investigations of

Cleveland and McGill as well as investigations by

perceptual psychologists.

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Bar Charts

The same

data are

represented

now with bar

charts. No-

tice that this

representa-

tion allows

a very clear

quantifica-

tion of the

differences

between the

ice-cream

types.

Blueberry Cherry Apple Boston Cream Other Vanilla Cream0.

000.

100.

200.

30

BlueberryCherryAppleBoston CreamOtherVanilla Cream

0.05 0.10 0.15 0.20 0.25 0.30

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Bar Charts

Strictly speaking bar charts can be used for drawing a summary of

either quantitative or qualitative data types. Qualitative data

types, such as nominal or ordinal variables, can be visualized using

bar charts.

The next data summary graphing techniques, Stem and Leaf Plots

and Histograms are used for quantitative variables.

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Stem and Leaf Plots

A similar technique used to graphically represent data is to use

stem and leaf plots. Take a look at the Healthy Breakfast data set.

This datafile contains nutritional information and grocery shelf

location for 77 breakfast cereals. Current research states that

adults should consume no more than 30 % of their calories in the

form of fat, they need about 50 grams (women) or 63 grams (men)

of protein daily, and should provide for the remainder of their

caloric intake with complex carbohydrates. One gram of fat

contains 9 calories and carbohydrates and proteins contain 4

calories per gram. A ”good” diet should also contain 20-35 grams

of dietary fiber.

Check out R code.

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Income level in $ frequency

0 – 1,000 211908788

1,000 – 2,000 423817576

2,000 – 3,000 635726364

3,000 – 4,000 847635152

4,000 – 5,000 1059543940

5,000 – 6,000 1059543940

6,000 – 7,000 1059543940

7,000 – 10,000 3178631820

10,000 – 15,000 5509628488

15,000 – 25,000 5509628488

25,000 – 50,000 1695270304

50,000 and over 211908788

Frequency Histograms

Consider the table of fami-

lies by income in the US in

1973 and corresponding per-

cents. In this table class in-

tervals include the left point,

but not the right point. It is

important to specify which of

the endpoints are included in

each class.

Notice that in this case class

intervals do not have the

same length.

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We can draw a frequency histogram for each of the income level

ranges specified by dividing the frequency counts by the total

number of families. However, the ranges are different widths, so

that the area of each block is NOT equally proportional to the

number of families with incomes in the corresponding class interval.

We really WANT the areas of each block to equally represent the

proportion of families within the income class interval, so instead

we use a density histogram.

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Density Histograms

Density histograms are sim-

ilar to frequency histogram

except heights of rectangles

are calculated by dividing

relative frequency by class

width. (frequency ÷ total

number of families ÷ class

width). Resulting rectangle

heights called densities, and

the vertical scale called ”den-

sity scale”.

Density Histogram for Income

Income in $1000

Den

sity

0 10 20 30 40 50

0.00

0.01

0.02

0.03

0.04

0.05

NOTE: I used total subjects in income data set = 211,908,788.

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IMPORTANT!

When comparing data sets with different sample sizes OR when

drawing a histograms with varying class interval widths, it is NOT

appropriate to compare raw frequency histograms. Why?

When sample sizes are different, density scale histograms are

BETTER.

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Drawing Density Histograms

Once the distribution table of percentages is available the next step

is to draw a horizontal axis specifying the class intervals.

Then we draw the blocks remembering that

In a density histogram

the areas of the blocks represent percentages

So, it is a mistake to set the heights of the blocks equal to the

percentages in the table. (that would be a relative frequency

histogram, which we’ll talk about next.)

To figure out the height of a block divide the percentage by the

class width of the interval.

The table needed to calculate the heights of the blocks looks like

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Income level in $ percent class width (in $1,000’s) height

0 – 1,000 1 1 1

1,000 – 2,000 2 1 2

2,000 – 3,000 3 1 3

3,000 – 4,000 4 1 4

4,000 – 5,000 5 1 5

5,000 – 6,000 5 1 5

6,000 – 7,000 5 1 5

7,000 – 10,000 15 3 5

10,000 – 15,000 26 5 5.2

15,000 – 25,000 26 10 2.6

25,000 – 50,000 8 25 .32

50,000 and over 1

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Distribution of family income in the US in 1973

income in $1000

perc

ent p

er $

1000

01

23

45

0 10 20 30 40 50

This is the resulting

histogram. The sum

of the areas of a den-

sity histogram adds to

1. Notice that the

class interval of incomes

above $50,000 has been

ignored.

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Vertical scale

What is the meaning of the vertical scale in a histogram?

Remember that the area of the blocks is proportional to the

percents. A high height implies that large chunks of area

accumulate in small portions of the horizontal scale.

This implies that the density of the data is high in the intervals

where the height is large. In other words, the data are more

crowded in those intervals.

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Another Example of a Density Histogram

Information is available

from 131 hospitals.

We show a histogram of

the average length of

stay measured in days for

each hospital.

The area of each block is

proportional to the num-

ber of hospitals in the cor-

responding class interval.

Histogram of the average length of stay in hospital

length of stay (days)

6 8 10 12 14 16 18 20

In this example all the intervals have the same length, so the

heights of the blocks give all the information about the number of

hospitals in each class.

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There are 7 class intervals corresponding to

• 6 to 8 days

• 8 to 10 days

• 10 to 12 days

• 12 to 14 days

• 14 to 16 days

• 16 to 18 days

• 18 to 20 days

Note that the class that corresponds to 14 to 16 days is empty and

that the class with the highest count of hospitals is the one of 8 to

10 days.

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Cross tabulation

In many situations we need to perform an exploratory analysis of

data to observe possible associations with a discrete variable. For

example, consider measuring the blood pressure of women and

divide them in two groups: one taking the contraceptive pill and

the other not taking it.

We can produce a table with the distribution of one group in one

column and the distribution of the other in another column. This

can be used to produce two histograms in order to make a visual

comparison of the the two groups.

The variable that is used for the cross-tabulation is usually referred

to as a covariable.

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blood pressure non users users

(mm) % %

under 100 8 6

100–110 20 12

110–120 31 26

120–130 19 22

130–140 13 17

140–150 6 11

150–160 2 4

over 160 1 2

women not using the pill

blood pressure (mm)

perc

ent p

er m

m

0.0

1.0

2.0

3.0

100 110 120 130 140 150 160

women using the pill

blood pressure (mm)

perc

ent p

er m

m

0.0

1.0

2.0

3.0

100 110 120 130 140 150 160

We observe that the histogram of pill users is slightly shifted to the

right, suggesting an increase in blood pressure among women

taking the pill. These are relative frequency histograms.

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Relative Frequency Histograms

women not using the pill

blood pressure (mm)

perc

ent p

er m

m

0.0

1.0

2.0

3.0

100 110 120 130 140 150 160

women using the pill

blood pressure (mm)

perc

ent p

er m

m

0.0

1.0

2.0

3.0

100 110 120 130 140 150 160

These are relative frequency

histograms, because the height

of the bars is the fraction of

times the value occurs e.g. the

frequency of value(s) ÷ number

of observations in the set. Rel-

ative frequency histograms are

also useful for comparing two

samples with different sample

sizes.

The Sum of all relative frequencies in a dataset is 1.

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Problems

Data from the 1990 Census produce the following for houses in the

New York City area that are either occupied by the owner or

rented out.

1. The owner-occupied percents add up to 99.9% and the

renter-occupied percents add up to 100.1%, why?

2. The percentage of one-room units is much larger for

renter-occupied housing. Is that because there is more

renter-occupied housing in total?

3. Which are larger on the whole: the owner-occupied units or the

renter-occupied units?

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Number of Rooms Owner Occupied Renter Occupied

1 2.0 9.2

2 3.8 12.9

3 11.9 32.5

4 14.5 26.5

5 16.7 12.5

6 22.3 4.8

7 11.7 1.0

8 6.5 0.3

≥ 9 10.5 0.4

Total 99.9 100.1

Number 785,120 1,782,459

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The answer to the first question is that there is rounding involved

in the calculation of the percentages.

As for the second question, the fact that we are taking percentages

accounts for the difference in totals, so a larger total of

renter-occupied units does not explain the difference. What seems

to be happening is that units for rent tend to be smaller than units

occupied by their owners. This is more clearly seen from the

comparison of the two histograms.

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1 2 3 4 5 6 7 8 9

Owner−occupied0

515

25

1 2 3 4 5 6 7 8 9

Renter−occupied

05

1525

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Average and spread in a histogram

A histogram provides a graphical description of the distribution of

a sample of data. If we want to summarize the properties of such a

distribution we can measure the center and the spread of the

histogram.

These two histograms cor-

respond to samples with

the same center.

The spread of the sample

on top is smaller than that

of the sample in the bot-

tom

Histogram of n1

n.1

Den

sity

−6 −4 −2 0 2 4 60.

000.

100.

200.

30

Histogram of n2

n.2

Den

sity

−6 −4 −2 0 2 4 6

0.00

0.10

0.20

0.30

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Average and median

In addition to summarizing a variable graphically to look for

patterns, it’s also useful to summarize it numerically.

The three most useful measures of center (or central tendency) are

the mean, the median (and other quantiles, or percentiles), and the

mode.

It turns out the the mean has the graphical interpretation of the

center of gravity of the data. If you visualize the histogram of a

variable as made of bricks that are sitting on a number line made of

plywood, which in turn is put on top of a saw-hours, the mean is

the place where the histogram would exactly balance.

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To obtain an estimate of the center of the distribution we can

calculate an average.

The average of a list of numbers equals their sum, divided by

how many they are

Thus, if 18; 18; 21; 20; 19; 20; 20; 20; 19; 20 are the ages of 10

students in this class, the average is given by

18 + 18 + 21 + 20 + 19 + 20 + 20 + 20 + 19 + 20

10= 19.5

In the hospital data that we considered in the previous class the

data corresponded to the average length of stay of patients in each

hospital in the survey. This means that the length of stay of all

patients in a given hospital were added and the sum divided by the

number of patients in that hospital.

(remember Summation Notation??)

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Average and median

The median of a column of numbers is found by sorting the data,

from smallest to highest, and finding the middle value in the list. If

the sorted list has an odd number of elements, then the median is

uniquely defined. If the sorted list has an even number of elements,

the median is the mean of the two middle values.

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histogram of rainfall in Guarico, Venezuela

mm

Den

sity

0 50 100 150 200 250

0.00

00.

005

0.01

00.

015

meanmedian

This histogram

corresponds to

the rainfall over

periods of 10 days

in an area of the

central plains of

Venezuela.

The average or

mean rainfall is

37.65 mm. We

observe that only

about 30% of the

observations are

above the average.

Notice that this histogram is not symmetric with respect to the

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average. The median of a histogram is the value with half the area

to the left and half to the right. The median and average of a

non-symmetric histogram are DIFFERENT.

histogram of rainfall in Guarico, Venezuela

mm

Den

sity

0 50 100 150 200 250

0.00

00.

005

0.01

00.

015

meanmedian

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A symmetric his-

togram will look

like this.

In this case 50% of

the data are above

the average.

Histogram of dat

dat

Den

sity

0 2 4 6 8 10

0.00

0.05

0.10

0.15

In a symmetric histogram the median and the average coincide.

By definition the median is the 50th percentile, although it’s also

useful sometimes to look at other percentiles, for example the 25th

percentile ( also called the first quartile) is the place where 1

4of the

data is to the left of that place.

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Average bigger thanmedian: long right tail

Average about the sameas median: symmetry

Average is smaller thanmedian: long left tail

The average is very sensitive to extreme observations, so when

dealing with variables like income or rainfall, that exhibit very long

tails, it is preferable to use the median as a measure of centrality.

The relationship between the average and the median determines

the shape of the tails of a histogram.

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Problem 1:According to the Department of Commerce, the mean

and median price of new houses sold in the United States in mid

1988 were 141, 200 and 117, 800. Which of these numbers is the

mean and which is the median? Explain your answer.

Problem 2: The number of deaths from cancer in the US has

risen steadily over time. In 1985, about 462,000 people died of

cancer, up from 331.000 deaths in 1970. A member of Congress

says that these numbers show that these numbers show that no

progress has been made in treating cancer. Explain how the

number of people dying of cancer could increase even if treatment

of the disease were improving. Then describe at least one variable

that would be a more appropriate measure of the effectiveness of

medical treatment for a potentially fatal disease.

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A measure of size

Consider the sample

0, 5,−8, 7,−3

How big are these five numbers? If we consider the average as a

measure of size then we obtain 0.2, which is a fairly small value

compared to 7. The trouble is that in the average large negative

quantities cancel large positive ones.

To avoid this problem we need a measure of size that disregards

signs. We proceed as follows:

1. square all values

2. Calculate the average of the resulting numbers

3. Take the root of the resulting mean.

This is called the root mean square size of the sample.

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For the previous data set we have

r.m.s. size =

√

02 + 52 + (−8)2 + 72 + (−3)2)

5=

√29.4 ≈ 5.4

We could have also considered the average disregarding the signs,

which amounts to

0 + 5 + 8 + 7 + 3

5= 4.6

Unfortunately the mathematical properties of this way of

measuring size are not as appealing as the ones of r.m.s.

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Spread

As we saw at the beginning of the lecture two samples can have the

same center and be scattered along their ranges in different ways.

To measure the way a sample is spread around its average we can

use the standard deviation, or SD.

The SD of a list of numbers measures how far away they are

from their average

Thus a large SD implies that many observations are far from the

overall average.

Most observations will be one SD from the average. Very few

will be more than two SDs away.

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Empirical Rule

SDs are a pain to compute by hand or with a calculator, and it’s

easy to make mistakes when doing so, so it’s good to have a simple

way to roughly approximate the SD of a list of numbers by looking

at its histogram.

• If you start at the mean and go one SD either way, you’ll

capture about 2

3of the data.

• Roughly 95% of the observations are within two SDs of the

average.

• Roughly 99% of the observations are within three SDs of the

average.

This statements are more accurate when the distribution is

symmetric.

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Generally, more data is better than less data because

more data mean less uncertainty (or smaller give or

take).

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Empirical Rule and the Cereal Data Example

Density Histogram

milligrams of sodium

Den

sity

0 50 100 150 200 250 300 350

0.00

00.

002

0.00

40.

006

0.00

8

Let’s look back at the ce-

real data example, which

has a mean of about 160

mg of sodium.

Using the empirical rule, what guess would you make for the SD?

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Goldilocks and the Cereal Data Example

If you guessed 20 mg, that would be too small, because 140-180

mg ought to be about 2

3of the data.

If you guessed 100 mg, that would be too large, because 60-260mg

is more than 2

3of the data.

If you guessed 80 mg, then 80-240 mg ought to be about 2

3of the

data, and 0-320mg would be about 95% of the data, which

looks about just right!

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Calculating the SD

To calculate the standard deviation of a sample follow the steps:

• Calculate the average

• Calculate the list of deviations from the average by taking the

difference between each datum and the average.

• Calculated the r.m.s. size of the resulting list.

SD = r.m.s. deviation from average.

Consider the list 20,10,15,15. Then

average =20 + 10 + 15 + 15

4= 15

The list of deviations is 5, -5, 0, 0. Then

SD =

√

52 + (−5)2 + 02 + 02

4=

√12.5 ≈ 3.5

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Using a calculator

Most scientific calculators will have a function to calculate the

average and the SD of a sample. The steps needed to obtain those

values vary from model to model.

The important fact is that most calculators do not produce the SD

as we have defined it here. They consider the sum of the squares of

the deviations over the total number of data minus one. So, if you

obtain the SD from your calculator (or spreadsheet), say SD∗, then

SD =

√

number of entries - one

number of entries× SD∗

Some calculators have both, SD and SD∗. Please read the manual

of your calculator regarding this fact.

Notice that the units of SD are the same as the original data. So if

the data were measured in years, SD is also in years.

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Problems

Problem 1: Both the following lists have the same average of 50.

Which one has the smaller SD and why? (Do no computations)

1. 50,40,60,30,70,25,75

2. 50,40,60,30,70,25,75,50,50,50

The second list has more entries at the average, so the SD is

smaller.

Repeat for the following two lists

1. 50,40,60,30,70,25,75

2. 50,40,60,30,70,25,75,99,1

The second list has two wild observations, 99 and 1, which are

away from the average, so the SD is larger.

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Problem 2: Consider the list of numbers

0.7 1.6 9.8 3.2 5.4 0.8 7.7 6.3 2.2 4.1

8.1 6.5 3.7 0.6 6.9 9.9 8.8 3.1 5.7 9.1

1. Without doing any arithmetic, guess whether the average is

around 1, 5 or 10.

Only three of the numbers are smaller than 1, none are bigger

than 10, so the average is around 5.

2. Without doing any arithmetic, guess whether the SD is around

1,3 or 6.

If the SD is 1, then the entries 0.6 and 9.9 are too far away

from the average. The entries are too concentrated around 5

for the SD to be 6. So the 3 is the most likely value.

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Problem 3: The usual method for determining heart rate is to

take the pulse and count the number of beats in a given time

period. The results are generally reported as beats per minute; for

instance, if the time period is 15 seconds, the count is multilied by

four. Take your pulse for two 15-sec. periods, two 30-sec. periods,

and two 1-minute periods. Convert the counts to beats per minute

and report the results. Which procedure do you think gives the

best results?? Why?

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