Chapter 2: Describing, Exploring & Comparing Data
Section
1
2
3
4
5
6
7
Title
Overview
Frequency Distributions
Visualizing Data
Measures of Center
Measures of Variation
Measures of Relative Standing
Exploratory Data Analysis (EDA)
Notes Pages
2
3–4
5–8
9 – 12
13 – 18
19 – 20
21
1
§2.1 Overview
There are 2 types of statistics: descriptive and inferential
It is the descriptive statistic that we will be discussing in this chapter. A descriptive
statistic describes the characteristics of data.
Following is a list of the 5 important characteristics:
1) The center of the data which a representative value that indicates where the
middle of the data lies. The most common descriptive statistic for measuring
center is the average.
2) The variation or scatter of the data is also important.
3) The distribution of the data tells the shape of the data. We are very familiar
and will become even more so with a distribution call the Normal – this is a
bell shaped curve, the type of distribution that class grades follow.
4) Outliers which are data points that lie outside the “normal” variation of the
data.
5) The change in characteristics of data over time.
Just like memorizing order of operations (Please Excuse My Dear Aunt Sally, PEMDAS)
or the order of the planets in our solar system (Matilda Visits Every Monday Just Stays
Until Noon Period) or the names of the Great Lakes (HOMES) , there is a mnemonic
device to help you remember these important characteristics of data:
Computer Viruses Destroy Or Terminate (CVDOT)
It is my belief as a statistics instructor that you can not understand a statistic until you
have learned how to calculate that statistic by hand, so even though we will be using
technology to calculate many statistics I will still require you to be able to find basic
descriptive statistics by hand for small data sets.
2
§2.2 Frequency Distributions
To help us in our study of the shape of data (distribution) we will be learning how to
compile a frequency distribution (table), a relative frequency distribution and a
cumulative frequency distribution. A frequency table will be created which will give
categories (not like the categories for ordinal data) and the number of experimental units (data
points) within each category. From these we will learn how to draw pictures that show us
distributions. Let’s go through an example to learn the process. Here is a summary of
the process to start:
Creating a Frequency Distribution (Table)
1. Decide upon the # of classes (the categories). There should be between 5 and 20 classes,
but if many have 1 or 0 data points then you have chosen too many. I will normally
tell you how many classes to use.
2. Find the class width (the range of data points in each class). This is the
(Maximum  Minimum)/ # of classes *
* You may need to round, usually up
3. Select a lower limit for the 1st class. This point does not have to be the lowest data
point, but it should make sense in terms of the data (no negatives if negatives don’t exist,
etc.). To this lower limit add the class width. This is the lower limit of the next
class, not the upper limit of the first class!!!! Continue until you’ve gotten all your
classes.
4. Since the last class has an upper limit that you have not found, be sure to fill it in, as
well as the other upper limits.
5. Make your counts as to the number of data points in each class.
Example:
The ages of people arrested for purse snatching are:
16, 41,25,21,30, 17, 29, 50, 30, & 39
Using 4 classes create a frequency distribution (table) for the data.
Classes:
Class Width:
Lower Limits:
Place Upper Limits:
Make counts:
3
Some places where students go wrong and that you should be cautious of are:
1) Are your classes mutually exclusive?
2) Did you include all classes even if one was zero?
3) Does the sum of frequencies add to the number of data points? This is really
important!
Next, we need to discuss the relative frequency distribution which is simply our
frequency distribution listed in terms of percent of the whole. To create a relative
frequency distribution you first create a frequency distribution and then divide each class
frequency by the total number of data points (sum of all frequencies).
Rel. Freq. = Class Freq.  Sum of Freq.
Example:
For the above data of purse-snatchers create the relative frequency
distribution.
Finally, we need to know how to create a cumulative frequency distribution. This is
just a running total that includes all classes below, but not any above. In a cumulative
frequency distribution the cumulative frequency in the last class must be the total number
of data points. We write the classes in a little different manner for a cumulative
frequency dist. – we use less than and use the lower limit of the next class.
Example:
For the purse-snatchers create a cumulative frequency dist.
Now for the real work, what does this all mean? Well, this is just a preview of how the
data is behaving. We are learning where the data stacks up, what percentage lies in
certain areas and at what point have we seen most of the data. A frequency distribution is
really just the first step in seeing the shape of the data, and we will be using it to draw an
actual pictorial representation in the next section. But, at this early stage we can begin by
comparing frequency distributions to other similar data or data over time and looking
for patterns – and really looking for patterns and finding out how mathematically
significant those patterns are is what statistics is all about!
4
§2.3 Visualizing Data
This section concerns itself with looking at pictures that show the shape of the data. We
will talk about histograms, which are types of graphs created from frequency & relative
frequency dist. We will also discuss line graphs of frequency (freq. polygon) and
cumulative frequency dist. (ogives). We then move to another visual called a stem and
leaf plot and its relative the dot plot. Finally we will discuss a scatter diagram, but will
leave the other visuals such as the Pareto Chart and Pie Chart and Time Series to the
book.
Let’s begin with the histogram. A histogram is constructed from a freq. or rel. freq. dist.
The horizontal axis consists of the scale of the data with either the midpoint of the class
(called class marks or midpoints) labeled or the class boundaries (the midpoint between upper and
lower limit) marked. The vertical scale is the frequency. Each class is shown with a
vertical bar. There are no gaps between the bars on a histogram! Everything must be
clearly marked!!
Creating a Histogram
1. Create a frequency or relative frequency table
2. Find the class boundaries (or the class marks)
3. Create an axis system with the class boundaries on the horizontal axis & frequencies
on the vertical axis (label clearly)
4. Create a bar the width of the class and the height of the frequency to represent each
class in the table.
Example:
Create a Histogram for the purse-snatcher data.
What does this do for us? Well, it shows you the shape of the data! We can see that as
age increases the number of people snatching purses goes down and that in-between age
we have the highest frequency. We see that it is a skewed data set; it is not normally
distributed (that bell-shaped curve that we see in grades).
If you create a relative frequency histogram you will not see any difference in shape.
You are seeing the exact same trends; it is just represented in terms of percentages
instead of raw numbers.
When we talk about a frequency polygon we are talking about the points that correspond
to the class marks (mid-point of the class) and the frequency of each class. From these points
we create a line graph. This line graph can be shown super-imposed on our histogram.
The line graph is nice for showing trends, because our eye follows the slope of the line.
As a result of seeing the slope we see increases and decreases according to class.
5
Example:
Draw the frequency polygon over the histogram above.
An ogive is a line plot for a cumulative freq. These are constructed starting with the class
boundaries and the cumulative frequencies. The major use would be to see the number of
values below a particular point in the data. It can also help us to see an overall trend in
data – where we are getting the most increase and where things level off. With some data
this is important (see the example in the book) and with others it is not as useful.
A stem-and-leaf plot gives us a nice visualization of the data shape, but unlike the
histogram, allows us to keep the original data. This is what makes it superior to a
histogram – there is no loss of the original data.
This type of plot works with a stem, which is based upon the tens, hundreds, etc. and the
leaf, which completes the number. You can think of this process like writing a number in
expanded form! For large data sets it is sometimes necessary to give each stem two
representations to spread the data out a little more (this does add some level of error to the
process and thus misrepresentation of the data is possible) , called an expanded stem-and-leaf. We
can also bring the data together a little more for smaller data sets creating a condensed
stem-and-leaf (the same problem with misrepresentation can occur).
Creating a Stem-and-Leaf Plot
1. Look over the data and decide upon stem and leaf (all below 100, use a stem of 10’s)
2. Write a table with stems on left and leaves on right (label the stem & leaf unit value)
3. Fill in the leaves to represent all the data points
Example:
Create a stem and leaf plot for the purse-snatchers data
16, 41, 25, 21, 30, 17, 29, 50, 30 & 39
Now, if you look at this on its side, you will see the nearly the same shape as the
histogram, but with the added benefit of still being able to see the original data. Another
advantage of the stem-and-leaf is that it sorts the data, which prepares us for finding
some other important descriptive statistics that require ordered or ranked data (such as
medians, quartiles & percentiles).
6
Another type of visual representation is the dot plot. The dot plot also keeps all the
original data, sorts it, but is worse at showing shape than the histogram and dot plot. The
dot plot is good at showing outliers, however.
Use a number line representative of the range of data values and place a dot for each
point just above its representative number on the number line to construct a dot plot. For
our purse-snatching data, a dot plot is not too informative! This is what it looks like.
55
45
30
15
The final type of visual that I want to discuss in class is called a scatter diagram or scatter
plot. This is very easy to construct and shows us trends that exist in the data. We will
use this type of plot for our excursion into regression in chapter 9. This type of visual
can only be used for a data set that has ordered pairs (two characteristics that are somehow
related, collected from the same people or at the same time& place, etc.) You’ll be happy to know
that I’m going to need a new data set to show this! All we have to do is construct ordered
pairs based upon paired data and plot those points on a coordinate system using
appropriately labeled axes for the independent and dependent variables.
Example:
The following data is the score of a math reasoning test and the
starting salary of the person scoring thusly on the test. Create a
scatter diagram based upon it.
X = score
Y = salary
78
89
85
93
92
99
100
100
85
84
7
Scatter plots can be used for noticing a relationship between two sets of data but we do
not want to make the mistake of causality. Just because you receive a high score on the
above test does not guarantee a high salary. There is a relationship (called a correlation), but
this does not mean that a high salary is caused by a great score, it could be that a person
had a lot of prior experience in the job and received a high salary for this reason. This
will be discussed at length in Chapter 9.
This chapter also includes discussions of the Pareto chart, pie charts and time-series
graphs. The Pareto chart is a bar graph for qualitative data where the bars are ordered in
descending height to help tell the story of what class (category) is most important. A pie
chart is the same type that you are familiar with from your everyday life and uses the
Relative Frequency Distribution to show which classes (categories) are the largest or most
influential. The time-series graphs are useful for seeing trends over time.
This section and the preceding are very helpful in showing how to visualize data
(CVDOT) without showing time progression. We can easily see the center of the data,
the variation, the distribution via a histogram and with the dot plot or stem and leaf we
can see outliers very easily. We can not see time trends with these techniques however!
8
§2.4 Measures of Center
Recall in §2.1 we talked about the 5 characteristics of data (CVDOT – Center, Variation,
Distribution, Outliers, Time). The 3 most important are center, variation and distribution.
In §2.2&2.3 we dealt with descriptive methods for showing the distribution (shape), and
now we need to discuss the center (a representative value).
There are 4 major measures of center, each measure being based upon different criteria
and some being more appropriate than others. These 4 major measures are:
Mean
Median
Mode
Midrange
The first in our discussion is the mean (also called the arithmetic mean). This measure is only
appropriate for ratio or interval data. It only has meaning for numbers that are somehow
related on a continuous ordered scale. It is an average. If we let “x” be a number from
the sample and “n” be the total number in the sample. In mathematics and science a 
(capital sigma – a Greek letter) is used to represent the sum. If we use a capital N then we are
talking about the total number in a population. We can compute the mean for a
population or for a sample. It is important to note that English letters are used to
describe a sample statistic and Greek letters are used to describe population parameters.
x (read x bar) is the sample mean
 (pronounced mu) is the population mean
x
/n
x
/N
It is important to note that population means are rarely known, and the sample mean is
usually used as an approximation.
Example:
A sample of six school buses in the Carlton District travel the
following distances each day:
14.2, 16.1, 7.9, 10.6, 11.2, 12.0
Find the sample mean.
Note:  would be the sum of all distances traveled by all Carlton District buses and then divided by the
total number of buses in the district. This example only uses a sample of 6 buses.
The next measure of center is the median. The median is the middle value of the ranked
data. The median can be found for interval, ratio and ordinal data, but not for nominal
data. The median is denoted as ~x, pronounced x-tilde. Here is the procedure for finding
the median. It is quite easy if there is an odd number of data points but when there are an
even number there is a slightly special procedure.
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Finding the Median
1. Rank the data in ascending order (a stem-and-leaf is nice for this)
2. a) If odd # there is a number that has an equal number above and an equal number
below it. For example if there are 15 points then the 8th is the median since there
are 7 above it and 7 below it. It is the middle of the data.
b) If there is an even number of data points the middle is between two points, so the
two points must be averaged. If there are 20 points then the middle is between the
10th and 11th
Example:
Find the median of the following ranked data
7.9, 10.6, 11.2, 12, 14.2, 16.1
The median can be used with the same types of data as the mean (ordinal and interval), so
why would we need the median instead of the mean? The answer is outliers. Outliers
can affect the mean and but they do not affect the median. So, once the distribution of
the data has been observed the decision as to which measure of center to use can be
made!
Note: The median is a better measure of center for highly skewed data or data which contains outliers.
The next measure of center we will discuss is the mode. The mode is the score that
appears most frequently. Ranking the data also helps to find the mode (hence the stemand-leaf plot has another use). The mode can be found for all for classifications of data,
but it is the only measure of center appropriate for nominal data! Data can be of three
types when considering the mode:
No Mode – Meaning that there is no data point is repeated
Bimodal – Meaning that there are 2 data points that appear with the greatest
frequency.
Multimodal – Meaning many data points appear with the greatest frequency
Example:
Find the mode(s) if one exists.
Confinement in days: 17, 19, 19, 4, 19, 21, 3, 21, 19
Hourly Incomes: 4, 9, 7, 16, 10
Test Scores: 81, 39, 100, 81, 69, 76, 42, 76
10
In conclusion, which measure of center is best used depends upon 2 things – first the
classification of data and second the presence of outliers. Mean is usually the measure of
center that is used but it is not the most appropriate when outliers are present due to the
strong influence that outliers can have on this measure.
The midrange of the data is not often that important except when looking the symmetry
of data and using it in comparison with other measures of center. It is nothing more than
taking the high and low data points’ sum and dividing it by two. Please see your text
more information (Ed. 9 p. 62-67, Essential Ed. 2 p. 60)
I am going to include an example for the mean of a distribution using a frequency table,
but we may not have time to discuss the example in class.
Example:
The following frequency table refers to a sample of purse snatchers. The
data points represent the ages of the sampled purse snatchers at the time of
their arrest.
Class (ages) Frequency
16 – 24
3
25 – 33
4
34 – 42
2
43 – 51
1
In order to calculate the approximate mean using a frequency table we will need to fill in
the following table.
f
Class
Mid-Point (x)
fx
16-24
3
(24+16)/2 = 20
203 = 60
25-33
4
34-42
2
43-51
1
n
fx 
x = fx
n
We will not discuss the weighted mean but you are expected to read the section
containing this information on p. 63-66 of Triola (9th ed) or p. 62 of Essentials ed. 2.
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Skewness is something that we need to talk about now that we have discussed the mean
and the median. It has to do with the distribution of the data with respect to the mean and
the median and mode and therefore must be left for discussion after the mean and the
median and mode. Skewness is a measure of symmetry. If a distribution extends more to
one side than the other of its central grouping then it is called skewed.
Left Skewed
(negative)
Mean & Median to the left
of the Mode
Outlying data is dragging
the mean & to less extent median
to the left of mode
Symmetric
MeanMedianMode
Data is in a central grouping
& outlying data is evenly spread
Right Skewed
(positive)
Mean & Median Right of
Mode
Outlying data is dragging
the mean & to less extent
median to right of mode
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§2.5 Measures of Variation
This section is about the measure of variation, our second characteristic of data. We will
be discussing the range and the standard deviation (variance), and how we can use these
measures to tell us about our data.
Range is very easy to define. It is how much the data varies from high to low. We find
the range by computing the difference between the high and the low data points (high 
low). The problem with the range is that it can be affected when there are outliers.
Outliers can make the data appear to have a much larger range than it actually does.
Example:
Find the range of the test scores:
81, 39, 100, 81, 69, 76, 42, 76
Note: If we look at the distribution of the data we see most of the data is 70 or above with 2 scores that are
very different. These 2 scores affect the range of the data drastically. If we compute the standard deviation
of these scores it will be less affected by the 2 very low scores, because most of the scores are near the top
end of the scale. This is what makes standard deviation better than range in showing variation.
Probably the most important measure of variation for ordinal and interval data is the
standard deviation. This is the measure of the variation about the mean. The standard
deviation is the square root of the variance, but it is used more often than the variance
because of the difficulty in interpreting the units associated with the variance (they are
squared, and the units of the mean are not). Let’s not be too quick to disregard the variance
however as it has a characteristic that is extremely important is more advanced statistics –
it is an unbiased estimator (it tends to be a good estimator of the actual population variance) . The
standard deviation of a population is called sigma and is represented by the Greek lower
case letter sigma (). The standard deviation of a sample is represented by the lower case
“s”. If we are talking about population variance it is 2 and sample variance s2. The
following is the formula for sample variance. Remember that the sample standard
deviation is the square root of the variance.
s2 =
nx2  (x)2
n(n  1)
=
 ( x  x )2
(n  1)
*Note: There are 2 ways to calculate the variance. The 1st formula is much easier than the 2nd with the use
of a scientific calculator. The 2nd formula, is easier when a scientific calculator is unavailable and the
calculation MUST be done entirely by hand. I should be noted that with a small data set, the 2nd is also a
nice formula to use, but the more data, the more cumbersome the formula becomes.
Example:
a)
The following are sampled finish times in a bike race (in minutes).
28, 22, 26, 33, 21, 23, 37, 24
Find the mean of the data.
13
b)
=
x
28
22
26
33
21
23
37
24
Complete the following table to calculate the variance using the 2nd
formula given above.
x2
x  x-bar
(x  x-bar)2
*Note: The calculation of any sample statistic should contain 1 more decimal than the original data.
Always maintain as many decimals as possible in the calculation process until the final answer is derived.
If you can’t possibly maintain all decimals, try to keep at least 4, preferably 6.
c)
Now use your calculator and the first formula to calculate the variance.
Start by inputting all data into the data register of a TI-83/84
(stateditenter data in L1) or with a simple scientific TI input data then use
+ until all data is entered. After inputting data on a TI-83/84
(statcalc1varstats2nd f(n)#1) or on simple TI 2nd f(n) left parenthesis.
d)
Find the standard deviation of the data by taking the square root of the
value found in b or c. Remember that those values should be the same!
e)
Interpretation of the standard deviation involves the mean. The std. dev.
in conjunction with the mean is used to give a range of values in which to
find the data. Nearly 70% of all symmetric data will fall within one
standard deviation of the mean (that is, 1s above and 1s below the mean; it
tells us how the data spreads out from the mean).
f)
If this data is considered to be symmetric, calculate the range of values
where you would expect to find about 70% of all bike times to be.
14
It should be noted that the formula for the population variance is slightly different than
the that of the sample variance. The following are the formulas for the population
variance.
2 =
Example:
 (x  )2
N
=
N x2  (x)2
N2
Six families live on Merimac Circle. The number of children in
each family is:
1, 2, 3, 5, 3, 4
Since we a using all the families on Merimac Circle this is
considered a population.
a)
On your own, calculate .
b)
On your own, complete the table below and calculate 2 based upon the
table using the 2nd formula above.
x2
x  
(x  )2
x
1
2
3
5
3
4
c)
On your own, calculate 2 using the 1st formula given above.
d)
Calculate the standard deviation of the population ().
*Note: In you should get 1.3 when rounded appropriately. On a calculator the pop. Std. dev. is given as
xn or simply x where as the sample std. dev. is given as sx or xn1.
15
The standard deviation can also be calculated using a frequency table. Although we have
not gone over a frequency table get, I will walk you through the calculation of the
standard deviation based upon a frequency table. We will need the following vocabulary,
which will also be used in frequency tables.
Class – The subdivisions of the data. All classes have equal widths. No class
should overlap another.
Class Width – The width of a class, found by subtracting the lower class limits of
two successive classes.
Lower Class Limit – The lowest point for which a data point is considered in a
class. The class limits should have the same # of decimal
places as the data.
Upper Class Limit – The highest point for which a data point is considered in a
class.
Class Boundaries – The points equidistant between successive classes. This is
found by adding a successive upper and lower limit and dividing
by 2, or by taking a successive upper and lower limit, subtracting
them and dividing by 2 and then adding this amount to each upper
limit to achieve the boundaries, or equivalently by subtracting that
amount from the lower limit to achieve the boundaries.
Class Midpoints (also referred to as Marks) – The point in the middle of each class.
This is found by adding the lower and upper
limits of the class and dividing by 2, or by
subtracting the upper and lower limits and
dividing by 2 and then adding that amount to
each lower limit.
Frequency – The number of data points in each class.
Example:
The following frequency table refers to a sample of purse snatchers. The
data points represent the ages of the sampled purse snatchers at the time of
their arrest.
Class (ages) Frequency
16 – 24
3
25 – 33
4
34 – 42
2
43 – 51
1
In order to calculate the approximate standard deviation using a frequency table we will
need to fill in the following table.
f
Class
Mid-Point (x)
x2
fx
f  x2
16-24
3
(24+16)/2 = 20
202 = 400
203 = 60
4003 = 1200
25-33
4
34-42
2
43-51
1
n
fx 
fx2 
16
After you have finished the table, use the values to calculate the variance of the sample
using the following formula:
s2 = nfx2  (fx)2
n(n  1)
*Note: This will not give the exact value of the variance, but as the data becomes more symmetric it will
give a better and better approximation. The actual variance of this data set is 11.9 years 2. Of course, I
don’t know what a squared year means, so it might be nice to put it in terms of a standard deviation!!! 
Even if we are unaware of all data in a data set and can make the assumption that the data
is approximately symmetric, we can find an approximate value to use in the place of the
standard deviation using something called: The Range Rule of Thumb. This comes
from the fact that for symmetric data nearly all the data (95%) will lie within two
standard deviation of the mean, and therefore that the maximum data point should be
approximately the mean plus twice the standard deviation and the minimum data point
should be approximately the mean minus twice the standard deviation (x-bar  2s).
Range Rule of Thumb –
Example:
s  range
4
For the bike racing data found on page 10 of these notes, find the
value that you would expect for the standard deviation using the
range rule of thumb. Compare this value to the actual standard
deviation. Based upon the fact that the actual and the approximate
should be pretty close if the data is symmetric, do you think the
data is symmetric (even knowing nothing about what symmetric really
means!)?
Now, let’s also test out that idea of the maximum and minimum data points, again
remembering that the data should be symmetric for the use of this approximation.
Example:
Bookstore sales receipts give a mean of $61.35 and a standard
deviation of $28.658. What would you expect the minimum and
maximum values in sales to be for the bookstore?
We have been “quoting” an important approximation in this last section called the
Empirical Rule. This rule has to do with symmetric, bell-shaped data and it tells us the
following:
68% of the data will fall within 1 standard deviation of the mean
95% of the data will fall within 2 standard deviations of the mean
99.7% of the data will fall within 3 standard deviations of the mean
17
Furthermore, we consider it to be usual for a data point to be within 2 standard deviations
of the mean and unusual to be beyond 2 standard deviations. Once we get beyond 3
standard deviations we consider data points to be possible outliers.
Example:
For a sample of blueberry cakes the mean weight was 500 g and
the std. dev. was 12 g. In what weight range would you usually
expect to find blueberry cakes from this population?
Now, we can also use the Empirical Rule in another way. We can use it to give
approximate percentages of the population that lie within certain ranges of data. We can
do this by using something called the z-score. The z-score tells us how many standard
deviations a data point is from the mean.
z-score –
Example:
x  x-bar
s
Using the Empirical Rule and the z-score, indicate what percentage
of the data you would expect to find between the values of 488 and
512 g.
Now, you might ask yourself, what if I know my data is not symmetric? In this case, we
have another rule of thumb called Chebyshev’s Theorem. It is a pretty simple
calculation and only works for values beyond 1 standard deviation from the mean. It tells
us what percentage of the data to expect with in a specified number of std. dev., based
upon that number of standard deviations (k).
% of the Data = 1  1/k2
Example:
If we assume that the data for the blueberry cakes is not
symmetric, what percent of the data should fall within 2 standard
deviation of the mean?
*Note: This value will not change, no matter what the data. It is also a smaller range than that given by the
Empirical Rule, since the data is not considered to be symmetric.
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§2.6 Measures of Relative Standing
This section considers measure of position which can help us to compare data. The most
useful comparison score is a standard score, or a z-score. We have already discussed this
measure of position in the last section. It measures the number of standard deviations
from the mean for a bell-shaped curve and can be used to compare different
measurements in an equivalent way. The larger the z-score the more unusual the value.
We expect usual z-scores to be within 2 standard deviations of the mean and unusual
values to be outside 2 standard deviations (an actual calculated value for standard deviation is
always preferred to an estimated value using the range rule of thumb) .
Z = x  x-bar
s
Example:
or
x  

For a population of patients with  = 60 years and  = 12 years,
would we consider 68 to be an unusual value? Why or why not?
The next important measures of position are percentiles and quartiles. A percentile is a
measure of the percentage of scores below a certain value. Quartiles are special
percentiles. The 1st quartile (Q1) is the same as the 25th percentile (P25). The 2nd quartile
(Q2) is the 50th percentile, also called the median (x-tilde). The 3rd quartile (Q3) is the
75th percentile (P75); it should be noted that the 3rd quartile is the same as P25, from the
upper end of the data.
To find the percentile represented by a data point, the following process should be
followed. The data must 1st be ordered and then:
Percentile = # of points below  100
Total
Example:
In the purse snatching data what is the percentile of 29?
Stem (x10) Leaf (x1)
1
67
2
159
3
009
4
1
5
0
Now, let’s consider what needs to be done to find a data point that represents a given
percentile. Again, the data must be ordered and then:
Indicator Function:
Lk = k  n
100
k = %tile, n = # of data pts.
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If L is whole number average that and next data point.
If L is a decimal, round up to the next whole and use that as the %tile.
Example:
For the following data that represents the decibels create by an
ordinary household item.
Stem (x1)
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
Leaf (x0.1)
0
45
7899
24478
26
9
4458
023568
0478
0167
068
0689
7
28
019
29
4
1
Find P10.
Find Q1
Find Q2
Find Q3
Another measure of position that we should discuss is the Interquartile Range. This is
referred to as the IQR. The IQR is nothing more than Q3  Q1. The interquartile range is
used to show where the bulk of the data resides. Seventy-five percent of the data should
lie in the IQR. As a result, the IQR can be used to pinpoint outliers as well. Anything
outside 3xIQR from the Q1 or Q3 should be considered as an outlier.
Example:
For the above data, should 77.1 be considered an outlier?
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§2.7 Exploratory Data Analysis (EDA)
Exploratory data analysis is the first step any Statistician takes when looking at a data set
for the first time. It is important to see trends in the data such as shape, center, and
variation. Usually the first exploratory analysis conducted is investigation of shape.
With consideration of shape and data type, the most appropriate measures of center and
variation can be calculated. Once a Statistician has conducted this exploratory analysis
they are prepared for further analysis of the data using methods to be discussed in the
remainder of the book.
Investigation of Shape
Historgrams/Stem&Leaf Plots/Dotplots/Boxplots
Measures of Center
Mean/Median/Modes
Measures of Variation
Variance/Standard Deviation/Range/IQR
Outlier Investigation
Minimum/Maximum/3xIQR beyond Q1&Q2
Of all the above exploratory analysis, the only one not discussed thus far has been the
Boxplot. The boxplot is a 5 number summary that shows center, variation, position,
spread and shape of the data, but it can not be discussed with the other graphical
representations of shape because it requires the use of the quartiles.
5 Number Summary
Minimum & Maximum
Q1, Q2 & Q3
A boxplot uses the 5 number summary in a scaled drawing where the maximum and
minimum are represented by a small marking (usually a vertical line), the 1st and 3rd quartiles
form a box with the median as a vertical divider of that box, and then “whiskers” are
drawn from the central box to the minimum and maximum. If there are outliers present,
they are sometimes represented by asterisks (especially in Minitab). The boxplot shows the
shape by showing where the “bulk” of the data lies in relation to the minimum and
maximum, as well as showing a “swaying” of the data based upon where the median lies
within the central box (think of the median as a fulcrum). Because the boxplot is a scaled
drawing, it can make a nice comparison tool for multiple data sets.
Example:
For the following data representing a sample of the measures of the
diameter (in feet) of Indian dwellings in Wisconsin, create a stem
and leaf plot to order the data, find the 5 number summary and
then draw a boxplot (I want labels on the boxplot indicating the 5 number
summary).
22, 24, 24, 30, 22, 20, 28, 30, 24, 34, 36, 15, 37
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