1

Chapter 2

Descriptive Statistics:

Organizing, Displaying and

Summarizing Data

Objectives

Student should be able to

• Organize data –

Tabulate data into frequency/relative frequency tables

• Display data graphically –

Qualitative data – pie charts, bar charts, Pareto Charts.

Quantitative data –

� Histograms, Stemplots, Dot plots and Boxplots.

� Describe the shape of the plot.

• Summarize data numerically –

Quantitative data only –

Measure of center – mean, median, midrange, and mode.

Measure of position – quartiles and percentiles.

Measure of spread/variation – range, variance, standard deviation, and inter-quartile range.

• Use TI graphing calculator to obtain statistics.

Organize Data

Tabulate data into frequency and

relative frequency Tables

Tabulate Qualitative Data

• Qualitative data values can be organized

by a frequency distribution

• A frequency distribution lists

– Each of the categories

– The frequency/counts for each category

Frequency Table

• A simple data set is

blue, blue, green, red, red, blue, red, blue

• A frequency table for this qualitative data

is

• The most commonly occurring color is

blue

3Red

1Green

4Blue

FrequencyColor

What Is A Relative Frequency?

• The relative frequencies are the proportions (or

percents) of the observations out of the total

• A relative frequency distribution lists– Each of the categories

– The relative frequency for each category

Relative frequency = Total

Frequency

2

Relative Frequency Table

• A relative frequency table for this

qualitative data is

• A relative frequency table can also be

constructed with percents (50%, 12.5%,

and 37.5% for the above table)

.375 (= 3/8)Red

.125 (= 1/8)Green

.500 (= 4/8)Blue

Relative FrequencyColor

Tabulate Quantitative Data

• Suppose we recorded number of customers

served each day for total of 40 days as below:

• We would like to compute the frequencies and

the relative frequencies

Frequency/Relative Frequency Table

The resulting frequencies and the relative

frequencies:

Display Data graphically

Qualitative data – Bar, Pareto, Pie Charts

Quantitative data – Histograms, Stemplots, Dot plots

Graphic Display for Qualitative

Data

Bar Charts, Pareto Charts, Pie

Charts

Bar and Pie Charts for Qualitative

Data

• Bar charts for our simple data (generated with Chart

command in Excel)

– Frequency bar chart

– Relative frequency bar chart

Note: Always label the axes, provide category and numeric scales,

and title when you present graphs.

Relative Frequency Bar Chart

0

0.1

0.2

0.3

0.4

0.5

0.6

Blue Green Red

Color

Relative Frequency

Frequency Bar Chart

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Blue Green Red

Color

Frequency

3

Pareto Charts

• A Pareto chart is a particular type of bar graph

• A Pareto differs from a bar chart only in that the

categories are arranged in order

– The category with the highest frequency is placed first

(on the extreme left)

– The second highest category is placed second

– Etc.

• Pareto charts are often used when there are many

categories but only the top few are of interest

Pareto Charts

Here shows a Pareto chart for the simple

data set:

0.125Green

0.375Red

0.5Blue

Relative FrequencyColor

Pareto Chart

0%

10%

20%

30%

40%

50%

60%

Blue Red Green

Color

Relative Frequency

Side-by-Side Bar Charts

• Use it to compare multiple bar charts.

• An example side-by-side bar chart comparing

educational attainment in 1990 versus 2003

Pie Charts

Pie Charts are used to display qualitative data. It shows the amount of data that belong to each category as a proportional part of a circle.

Notice that Bar charts show the amount of data that belong to each category as a proportionally sized rectangular area.

Pie Chart

Blue, 50%

Red, 38%

Green, 13%

Pie Charts

• Another example of a pie chart

Summary

• Qualitative data can be organized in

several ways

– Tables are useful for listing the data, its

frequencies, and its relative frequencies

– Charts such as bar charts, Pareto charts, and

pie charts are useful visual methods for

organizing data

– Side-by-side bar charts are useful for

comparing multiple sets of qualitative data

4

Graphic Display Quantitative

Data

Histograms, Stemplots, Dot Plots

Histogram

Histogram is a bar graph which represents a frequency distribution of a quantitative variable. It is a term used only for a bar graph of quantitative data. A histogram is made up of the following components:

1. A title, which identifies the population of interest

2. A vertical scale, which identifies the frequencies or relative frequency in the various classes

3. A horizontal scale, which identifies the variable x. Values or ranges of values may be labeled along the x-axis. Use whichever method of labeling the axis best presents the variable.

When you make a graph, make sure you label (give descriptions to) both axes clearly, and give a title for the graph too.

Histogram for discrete Quantitative

data• Example of histograms for discrete data

– Frequencies

– Relative frequencies

Note: The term “histogram” is used only for a bar graph to

summarize quantitative data. The bar chart for qualitative data

can not be called a histogram. Also, there are no gaps between

bars in a histogram.

Categorize/Group Continuous

Quantitative Data

• Continuous type of quantitative data cannot be

put directly into frequency tables since they do

not have any obvious categories

• Categories are created using classes, or

intervals/ranges of numbers

• The continuous data is then put into the classes

Categorize/Group Continuous

Quantitative Data• For ages of adults, a possible set of classes is

20 – 29

30 – 39

40 – 49

50 – 59

60 and older

• For the class 30 – 39

– 30 is the lower class limit

– 39 is the upper class limit

• The class width is the difference between the upper class limit and the lower class limit

• For the class 30 – 39, the class width is

40 – 30 = 10

(The difference between two adjacent lower class limits)

• The class midpoint = Average of the lower limits for the two adjacent classes

Categorize/Group Continuous

Quantitative Data

• All the classes should have the same

widths, except for the last class

• The class “60 and above” is an open-

ended class because it has no upper limit

• Classes with no lower limits are also called

open-ended classes

5

Categorize/Group Continuous

Quantitative Data

• The classes and the number of values in

each can be put into a frequency table

• In this table, there are 1147 subjects

between 30 and 39 years old

11060 and older

49350 – 59

109040 – 49

114730 – 39

53320 – 29

Number

(frequency)

Age

Categorize/Group Continuous

Quantitative Data

• Good practices for constructing tables for

continuous variables

– The classes should not overlap

– The classes should not have any gaps between them

– The classes should have the same width (except for

possible open-ended classes at the extreme low or

extreme high ends)

– The class boundaries should be “reasonable”

numbers

– The class width should be a “reasonable” number

Histogram for continuous

Quantitative data• Just as for discrete data, a histogram can be created from the

frequency table

• Instead of individual data values, the categories are the classes –

the intervals of data

• You can label/scale the bars with the lower class limits or class

midpoints.

Stemplots

• A stem-and-leaf plot ( or simply Stemplot) is a

different way to represent data that is similar to a

histogram

• To draw a stem-and-leaf plot, each data value

must be broken up into two components

– The stem consists of all the digits except for the right

most one

– The leaf consists of the right most digit

– For the number 173, for example, the stem would be

“17” and the leaf would be “3”

Example of a Stemplot

• In the stem-and-leaf plot below

– The smallest value is 56

– The largest value is 180

– The second largest value is 178

Stemplots Construction

• To draw a stem-and-leaf plot

– Write all the values in ascending order

– Find the stems and write them vertically in ascending

order

– For each data value, write its leaf in the row next to its

stem

– The resulting leaves will also be in ascending order

• The list of stems with their corresponding leaves

is the stem-and-leaf plot

6

Modification to Stemplots

• Modifications to stem-and-leaf plots

– Sometimes there are too many values with the same stem … we would need to split the stems (such as having 10-14 in one stem and 15-19 in another)

– If we wanted to compare two sets of data, we could draw two stem-and-leaf plots using the same stem, with leaves going left (for one set of data) and right (for the other set) – a side-by-side stem plot

Dot Plots

• A dot plot is a graph where a dot is placed

over the observation each time it is

observed

• The following is an example of a dot plot

Shapes of Plots for Quantiative

Data• The pattern of variability displayed by the data of a variable is called

distribution. The distribution displays how frequent each value of the variable occurs.

• A useful way to describe a quantitative variable is by the shape of its distribution

• Some common distribution shapes are

– Uniform

– Bell-shaped (or normal)

– Skewed right

– Skewed left

– Bimodal

Note: We are not concerned about the shapes of the plots for qualitative data, because there is no particular order arrangement for the categories of the nominal data. Once we change the order, the shape of the graph will be changed.

Uniform Distribution

• A variable has a uniform distribution when

– Each of the values tends to occur with the

same frequency

– The histogram looks flat

Normal Distribution

• A variable has a bell-shaped (normal)

distribution when

– Most of the values fall in the middle

– The frequencies tail off to the left and to the

right

– It is symmetric

Right-skewed Distribution

• A variable has a skewed right distribution when

– The distribution is not symmetric

– The tail to the right is longer than the tail to the left

– The arrow from the middle to the long tail points right

In Other words: The direction of skewness is determined by the side of distribution

with a longer tail. That is, if a distribution has a longer tail on its right side, it is

called a right-skewed distribution.

Right

7

Left-skewed Distribution

• A variable has a skewed left distribution when

– The distribution is not symmetric

– The tail to the left is longer than the tail to the right

– The arrow from the middle to the long tail points left

Left

Bimodal Distribution

• There are two peaks/humps or highest points in the distribution.

• Often implies two populations are sampled.

The graph below shows a bimodal distribution for body mass. It implies that data come from two populations, each with its own separate average. Here, one group has an average body mass of 147 grams and the other has a average body mass of 178 grams.

Summary

• Quantitative data can be organized in several ways

– Histogram is the most used graphical tool.

– Histograms based on data values are good for discrete data

– Histograms based on classes (intervals) are good for continuous data

– The shape of a distribution describes a variable … histograms are useful for identifying the shapes

Summarize data numerically

Measure of Center, Spread, and

Position

Measure of Center

Mean, Median, Mode, Midrange

Measures of Center

• Numerical values used to locate the

middle of a set of data, or where the data

is most clustered

• The term mean/average is often

associated with the measure of center of a

distribution.

8

Mean

• An arithmetic mean

• For a population … the population mean

– Is computed using all the observations in a population

– Is denoted by a Greek letter µ ( called mu)

– Is a parameter

• For a sample … the sample mean

– Is computed using only the observations in a sample

– Is denoted (called x bar)

– Is a statistic

Note: We usually cannot measure µ (due to the size of the population) but would like to estimate its value with a sample

mean

x

x

Formula for Means• The sample mean is the sum of all the values divided by the size of

the sample, n:

• The population mean is the sum of all the values divided by the size

of the population, N:

)...(11

21 ni xxxn

xn

x +++== ∑

)...(11

21 Ni xxxN

xN

+++== ∑µ

Note: is called “summation”, means summing all values. ∑

It is a short-cut notation for adding a set of numbers.

Example� Example:The following sample data represents the number

of accidents in each of the last 6 years at a dangerous

intersection. Find the mean number of accidents: 8, 9, 3, 5,

2, 6, 4, 5:

x = + + + + + + + =1

88 9 3 5 2 6 4 5 525( ) .Solution:

� In the data above, change 6 to 26:

Note: The mean can be greatly influenced by outliers (extremely large or

small values)

x = + + + + + + + =1

88 9 3 5 2 26 4 5 7 75( ) .Solution:

Median

• The median denoted by M of a variable is the “center”. The median splits the data into halves

• When the data is sorted in order, the median is the middle value

• The calculation of the median of a variable is slightly different depending on

– If there are an odd number of points, or

– If there are an even number of points

How to Obtain a Median?

• To calculate the median of a data set– Arrange the data in order

– Count the number of observations, n

• If n is odd– There is a value that’s exactly in the middle

– That value is the median

• If n is even– There are two values on either side of the exact middle

– Take their mean to be the median

Example

• An example with an odd number of observations

(5 observations)

• Compute the median of

6, 1, 11, 2, 11

• Sort them in order

1, 2, 6, 11, 11

• The middle number is 6, so the median is 6

9

Example

• An example with an even number of observations (4 observations)

• Compute the median of

6, 1, 11, 2

• Sort them in order

1, 2, 6, 11

• Take the mean of the two middle values

(2 + 6) / 2 = 4

• The median is 4

Quick Way to Locate Median

1. Rank the data (Suppose, the sample size is .)

2. Find the position of the median (counting from

either end) using the formula:

2

1+=n

i

Then, the median is the ith smallest value.

n

Example 1

Suppose we want to find the median of the data set

4, 8, 3, 8, 2, 9, 2, 11, 3,

1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11

2. Find the position of the median using the formula:

For the data given, n is 9 (because the size of the

sample is 9, that is, there are 9 data values given),

so the median position is

The median is the 5th smallest or 5th largest value, which

is 4.

2

1+n

52

19=

+

Example 2

Consider this data set 4, 8, 3, 8, 2, 9, 2, 11, 3, 15

1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11, 15

2. Find the position of the median using the formula:

For the data given, n is 10 (because the size of the

sample is 10, that is, there are 10 data values given),

so the median position is

The median is the 5.5th smallest or largest value. In other words,

it is in the middle of the 5th and 6th smallest or largest values. Since the 5th value is 4 and the 6th value is 8. We average out 4 and 8, so the median is 6.

2

1+n

5.52

110=

+

Mode

The mode of a variable is the most frequently occurring value.

For instance, Find the mode of the data

6, 1, 2, 6, 11, 7, 3

Since the data contain 6 distinct values:

1, 2, 3, 6, 7, 11

and, the value 6 occurs twice, all the other values occur

only once, so the mode is 6

Note: If two or more values in a sample are tied for the highest frequency (number of occurrences), there is no mode

midrange=+L H

2

Midrange

Another useful measure of the center of the distribution

is Midrange, which is the number exactly midway

between a lowest value data L and a highest value

data H. It is found by averaging the low and the

high values:

10

Comparing mean and Median

• The mean and the median are often

different

• This difference gives us clues about the

shape of the distribution

– Is it symmetric?

– Is it skewed left?

– Is it skewed right?

– Are there any extreme values?

Mean and Median

• Symmetric – the mean will usually be close to the

median

• Skewed left – the mean will usually be smaller than the

median

• Skewed right – the mean will usually be larger than the

median

Symmetric Distribution

• If a distribution is symmetric, the data values above and

below the mean will balance

– The mean will be in the “middle”

– The median will be in the “middle”

• Thus the mean will be close to the median, in general,

for a distribution that is symmetric

Left-skewed Distribution

• If a distribution is skewed left, there will be some data

values that are larger than the others

– The mean will decrease

– The median will not decrease as much

• Thus the mean will be smaller than the median, in

general, for a distribution that is skewed left

Right-skewed Distribution

• If a distribution is skewed right, there will be some data

values that are larger than the others

– The mean will increase

– The median will not increase as much

• Thus the mean will be larger than the median, in general,

for a distribution that is skewed right

Mean and Median

If one value in a data set is extremely different from the others?

For instance, if we made a mistake and

6, 1, 2

was recorded as

6000, 1, 2

• The mean is now ( 6000 + 1 + 2 ) / 3 = 2001

• The median is still 2

• The median is “resistant to extreme values” than the mean.

11

Round-off Rule

� When rounding off an answer, a common rule-of-thumb is to keep one more decimal place in the answer than was present in the original data

� To avoid round-off buildup, round off only the final answer, not intermediate steps

Measure of Spread

Range, Variance, Standard Deviation

Measures of Spread/Dispersion

• Measures of central tendency alone cannot completely characterize a set of data. Two very different data sets may have similar measures of central tendency.

• Measures of dispersion are used to describe the spread, or variability, of a distribution

• Common measures of dispersion: range, variance, and standard deviation

Range

• The range of a variable is the largest data value minus

the smallest data value

• Compute the range of

6, 1, 2, 6, 11, 7, 3, 3

• The largest value is 11

• The smallest value is 1

• Subtracting the two … 11 – 1 = 10 … the range is 10

Note: Please do not confused the range with the midrange

which is a measure for the center of data distribution

Range

• The range only uses two values in the data set –

the largest value and the smallest value

• The range is affected easily by extreme values

in the data. (i.e., not resistant to outliers)

• If we made a mistake and

6, 1, 2

was recorded as

6000, 1, 2

• The range is now ( 6000 – 1 ) = 5999

Deviations From The Mean

• The variance is based on the deviation from the mean

– ( xi – µ ) for populations

– ( xi – ) for samples

• Deviation may be positive or negative depending on if value is

above the mean or below the mean. So, the sum of all deviations

will be zero. To avoid the cancellation of the positive deviations

and the negative deviations when we add them up, we square the

deviations first:

– ( xi – µ )2 for populations

– ( xi – )2 for samples

x

x

12

Population Variance

• The population variance of a variable is the average of these squared deviations, i.e. is the sum of these squared deviations divided by the number in the population

• The population variance is represented by σ2 (namely sigma square)

Note: For accuracy, use as many decimal places as allowed by your calculator during the calculation of the squared deviations, if the average is not a whole number.

N

xxx

N

xNi

22

2

2

1

2)(...)()()( µµµµ −++−+−

=−∑

Example

• Compute the population variance of

6, 1, 2, 11

• Compute the population mean first

µ = (6 + 1 + 2 + 11) / 4 = 5

• Now compute the squared deviations

(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36

• Average the squared deviations

(16 + 9 + 1 + 36) / 4 = 15.5

• The population variance σ2 is 15.5

Sample Variance

• The sample variance of a variable is the average

deviations for the sample data, i.e., is the sum of these

squared deviations divided by one less than the number

in the sample

• The sample variance is represented by s2

Note: we use n – 1 as the devisor.

1

)(...)()(

1

)( 22

2

2

1

2

−

−++−+−=

−

−∑n

xxxxxx

n

xxNi

Example

• Compute the sample variance of

6, 1, 2, 11

• Compute the sample mean first

= (6 + 1 + 2 + 11) / 4 = 5

• Now compute the squared deviations

(1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36

• Average the squared deviations

(16 + 9 + 1 + 36) / 3 = 20.7

• The sample variance s2 is 20.7

( )

1

2

2

2

−

−=

∑∑

nn

xx

s

Computational Formulas for the

Sample Variance� A shortcut (a quick way to compute) formula for the sample

variance: ( because you do not need to compute all the deviations

from the mean.)

∑ 2x is the sum of the squars of each data value.

is the square of the sum of all data values.

For the above example, ,

( )2∑ x

16211216 22222 =+++=∑ x ( ) 400)11216( 22=+++=∑ x

7.2014

4

400162

2 =−

−=S

Compare Population and Sample

Variances

• Why are the population variance (15.5) and the sample variance (20.7) different for the same set of numbers?

• In the first case, { 6, 1, 2, 11 } was the entire population (divide by N)

• In the second case, { 6, 1, 2, 11 } was just a sample from the population (divide by n – 1)

• These are two different situations

13

Why Population and Sample

Variances are different?

• Why do we use different formulas?

• The reason is that using the sample mean is not quite as accurate as using the population mean

• If we used “n” in the denominator for the sample variance calculation, we would get a “biased” result

• Bias here means that we would tend to underestimate the true variance

Standard Deviation

• The standard deviation is the square root of the variance

• The population standard deviation

– Is the square root of the population variance (σ2)

– Is represented by σ

• The sample standard deviation

– Is the square root of the sample variance (s2)

– Is represented by s

Note: Standard deviation can be interpreted as the average deviation of the data. It has the same measuring unit as the original data ( e.g. inches). The variance has a squared unit (e.g. inches2).

Example

• If the population is { 6, 1, 2, 11 }

– The population variance σ2 = 15.5

– The population standard deviation σ =

• If the sample is { 6, 1, 2, 11 }

– The sample variance s2 = 20.7

– The sample standard deviation s =

• The population standard deviation and the sample standard deviation apply in different situations

9.35.15 =

5.47.20 =

• In a grouped frequency distribution, we use the

frequency of occurrence associated with each

class midpoint

• In an ungrouped frequency distribution, use the

frequency of occurrence, f, of each observation

xxf

f=∑∑

( )

s

x fxf

f

f

2

2

2

1=

−

−

∑∑∑

∑

Compute mean and Variance for A

Frequency Distribution

To calculate the mean, variance for a set of

sample data:

Grouped Data

• To compute the mean, variance, and standard deviation for grouped

data

– Assume that, within each class, the mean of the data is equal to

the class midpoint (which is an average of two adjacent lower

lass limits.)

– Use the class midpoint as an approximated value for all data in

the same class, since their actual values are not provided.

– The number of times the class midpoint value is used is equal to

the frequency of the class

• For instance, if 6 values are in the interval [ 8, 10 ] , then we assume

that all 6 values are equal to 9 (the midpoint of [ 8, 10 ]

Example of Grouped Data

• As an example, for the following frequency table,

we calculate the mean as if– The value 1 occurred 3 times

– The value 3 occurred 7 times

– The value 5 occurred 6 times

– The value 7 occurred 1 time

7531Midpoint

1673Frequency

6 – 7.94 – 5.92 – 3.90 – 1.9Class

14

Example of Grouped Data

The calculation for the mean would be

17

75555553333333111 ++++++++++++++++

17

)17()65()73()31( ×+×+×+×Or

Which follows the formula

7531Midpoint

1673Frequency

6 – 7.94 – 5.92 – 3.90 – 1.9Class

∑∑

=f

xfX

= 3.6

Example of Grouped Data

Since

the sample size = 171673 =+++=∑ f

the Sum of squared values = 2651765733122222 =×+×+×+×=∑ fx

the square of the sum = ( ) 372161)17657331(222==×+×+×+×=∑ fx

Follow the short-cut formula for the sample variance, we obtain

882.216

88235.218265

117

17

3721265

2 =−

=−

−=S

7.1882.2 ==S

the sample variance

the sample standard deviation

Summary

• The mean for grouped data

– Use the class midpoints

– Obtain an approximation for the mean

• The variance and standard deviation for

grouped data

– Use the class midpoints

– Obtain an approximation for the variance and

standard deviation

� Example: A survey of students in the first grade at a local school asked for the number of brothers and/or sisters for each child. The results are summarized in the table below. Here, we see 15 students responded o sibling, 17 students responded 1 sibling, etc. Total number of students in this survey is 62, which is .

� Find 1) the mean, 2) the variance, and 3) the standard deviation:

Example of Ungrouped Data

s2

2239 93

6262 1

163=−

− =

( )

.2) s= =163 128. .3)x= =93 62 15/ .1)

Solutions:First:

172352

17462010

1245

62 93Sum:

15 00x f x f x f2

17928050239

0

∑= fn

Measure of Position

Percentiles, Quartiles

Measures of Position

• Measures of position are used to describe the

relative location of an observation within a data

set.

• Quartiles and percentiles are two of the most

popular measures of position

• Quartiles are part of the 5-number summary

15

Percentile

• The median divides the lower 50% of the data

from the upper 50%

• The median is the 50th percentile

• If a number divides the lower 34% of the data

from the upper 66%, that number is the 34th

percentile

Quartiles

• Quartiles divide the data set into four equal parts

• The quartiles are the 25th, 50th, and 75th percentiles

– Q1 = 25th percentile

– Q2 = 50th percentile = median

– Q3 = 75th percentile

• Quartiles are the most commonly used percentiles

• The 50th percentile and the second quartile Q2 are both other ways

of defining the median

How to Find Quartiles?

1. Order the data from smallest to largest.

2. Find the median Q2.

3. The first quartile (Q1) is then the median of the lower half of the data; that is, it is the median of the data falling below the median (Q2) position (and not including Q2).

4. The third quartile (Q3) is the median of the upper half of the data; that is, it is the median of the data falling above the Q2 position (not including Q2).

Note: Excel has a set of different rules to compute these quartiles than the TI graphing calculator which will follow the rules stated above. So, different software may give different quartiles, particularly if the sample size is an odd-numbed. However, for a large data set, the values are often not much different. In our class, we will only follow the rules stated here.

1) Median= Q2 = the average of the 10th and 11th smallest values = (6.4+6.7)/2 =6.55

5.6 5.6 5.8 5.9 6.0

6.0 6.1 6.2 6.3 6.4

6.7 6.8 6.8 6.8 6.9

7.0 7.3 7.4 7.4 7.5

ExampleThe following data represents the pH levels of a random sample of

swimming pools in a California town. Find the three quartiles.

2) The first quartile = Q1 = the median of the 10 values below the median

= the average of the 5th and 6th smallest values = (6.0+6.0)/2 = 6.0

3) The third quartile =Q3 = the median of the 10 values above the median

= the average of the 15th and 16th smallest values = (6.9+7.0)/2 = 6.95

Solutions:

Outliers

• Extreme observations in the data are referred to as outliers

• Outliers should be investigated

• Outliers could be

– Chance occurrences

– Measurement errors

– Data entry errors

– Sampling errors

• Outliers are not necessarily invalid data

How To Detect Outliers?

• One way to check for outliers uses the quartiles

• Outliers can be detected as values that are

significantly too high or too low, based on the

known spread

• The fences used to identify outliers are

– Lower fence = LF = Q1 – 1.5 × IQR

– Upper fence = UF = Q3 + 1.5 × IQR

• Values less than the lower fence or more than

the upper fence could be considered outliers

16

Example

• Is the value 54 an outlier?

1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54

• Calculations

– Q1 = (4 + 7) / 2 = 5.5

– Q3 = (27 + 31) / 2 = 29

– IQR = 29 – 5.5 = 23.5

– UF = Q3 + 1.5 × IQR = 29 + 1.5 × 23.5 = 64

• Using the fence rule, the value 54 is not an outlier

Another Measure of the

Spread

Inter-quartile range (IQR)

Inter-quartile Range (IQR)

• The inter-quartile range (IQR) is the difference

between the third and first quartiles

IQR = Q3 – Q1

• The IQR is a resistant measurement of spread.

Its value will not be affected easily by extremely

large or small values in a data set, since IQR

covers only the middle 50% of values.)

Another Graphical Tool to

Summarize Data

Five-number Summary

&

Boxplot

Five-number Summary

• The five-number summary is the collection of

– The smallest value

– The first quartile (Q1 or P25)

– The median (M or Q2 or P50)

– The third quartile (Q3 or P75)

– The largest value

• These five numbers give a concise description of the distribution of a variable

Why These Five Numbers?

• The median

– Information about the center of the data

– Resistant measure of a center

• The first quartile and the third quartile

– Information about the spread of the data

– Resistant measure of a spread

• The smallest value and the largest value

– Information about the tails of the data

17

Example

• Compute the five-number summary for the ordered data:

1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54

• Calculations– The minimum = 1

– Q1 = P25, Q1 = 7

– M = Q2 = P50 = (16 + 19) / 2 = 17.5

– Q3 = P75 = 27

– The maximum = 54

• The five-number summary is1, 7, 17.5, 27, 54

Boxplot

• The five-number summary can be illustrated using a graph called the boxplot

• An example of a (basic) boxplot is

• The middle box shows Q1, Q2, and Q3

• The horizontal lines (sometimes called “whiskers”) show the minimum and maximum

How to draw A Boxplot?

To draw a (basic) boxplot:

1. Calculate the five-number summary

2. Draw & scale a horizontal number line which will cover all the data from the minimum to the maximum

3. Mark the 5 numbers on the number line according to the scale.

4. Superimpose these five marked points on some distance above the lines.

5. Draw a box with the left edge at Q1 and the right edge at Q3

6. Draw a line inside the box at M = Q2

7. Draw a horizontal line from the Q1 edge of the box to the minimum and one from the Q3 edge of the box to the maximum

Example

• To draw a (basic) boxplot

Draw the middle box

Draw in the median

Draw the minimum and maximum

Voila!

A Modified Boxplot

• An example of a more sophisticated boxplot is

• The middle box shows Q1, Q2, and Q3

• The horizontal lines (sometimes called

“whiskers”) show the minimum and maximum

• The asterisk on the right shows an outlier

(determined by using the upper fence)

How To Draw A Modified Boxplot?

To draw a modified boxplot1. Draw the center box and mark the median, as before

2. Compute the upper fence and the lower fence

3. Temporarily remove the outliers as identified by the upper fence and the lower fence (but we will add them back later with asterisks)

4. Draw the horizontal lines to the new minimum and new maximum (These are the minimum and maximum within the fence)

5. Mark each of the outliers with an asterisk

Note: Sometimes, data contain no outliers. You will obtain a basic boxplot.

18

Example

• To draw this boxplot

Draw the middle box and the median

Draw in the fences, remove the outliers (temporarily)

Draw the minimum and maximum

Draw the outliers as asterisks

Interpret a Boxplot

• The distribution shape and boxplot are related

– Symmetry (or lack of symmetry)

– Quartiles

– Maximum and minimum

• Relate the distribution shape to the boxplot for

– Symmetric distributions

– Skewed left distributions

– Skewed right distributions

Symmetric Distribution

Q1 M Q3Q1 M Q3 Q1 M Q3Min Max

The median line is in the

center of the box

Q1 is equally far from the median as

Q3 is

The left whisker is equal

to the right whisker

The min is equally far from

the median as the max is

BoxplotDistribution

Left-skewed Distribution

Q1 MQ3Q1 MQ3 Min MaxQ1 MQ3Min MaxQ1 MQ3

The left whisker is longer

than the right whisker

The min is further from the median

than the max is

The median line is to the

right of center in the box

Q1 is further from the median than Q3

is

BoxplotDistribution

Right-skewed Distribution

Q1M Q3 Q1M Q3Min MaxQ1M Q3Min MaxQ1M Q3

The median line is to the

left of center in the box

Q1 is closer to the median

than Q3 is

The left whisker is shorter

than the right whisker

The min is closer to the median

than the max is

BoxplotDistribution

Side-by-side Boxplot

• We can compare two distributions by

examining their boxplots

• We draw the boxplots on the same

horizontal scale

– We can visually compare the centers

– We can visually compare the spreads

– We can visually compare the extremes

19

Example

Comparing the “flight” with the “control” samples

CenterSpread

Summary

• 5-number summary

– Minimum, first quartile, median, third quartile

maximum

– Resistant measures of center (median) and spread

(interquartile range)

• Boxplots

– Visual representation of the 5-number summary

– Related to the shape of the distribution

– Can be used to compare multiple distributions

Using Technology for

Statistics

Instruction for TI Graphing

Calculator

Entering Data into TI Calculator

Enter data in lists: Press STAT then choose EDIT menu. (We’ll denote the sequence of the key strokes by STAT �EDIT). Entering data one by one (press Return after each entry) under a blank column which represents a variable (a list).

Note:

1. Clear a list: on EDIT screen, use the up arrow to place the cursor on the list name, press CLEAR, then ENTER (that is, CLEAR�ENTER). You need to always clear a list before entering a new set of data into the list.Warning! Pressing the DEL key instead of CLEAR will delete the list from the calculator. You can get it back with the INS key. See Insert a new list below.

2. List name: there are six built-in lists, L1 through L6, and you can add more with your own names. You can get the L1 symbol by pressing the 2ND key, then 1 key [ 2nd 1 ] .(The instruction in the brackets shows the sequence of keys you need to press, here, you press 2ND key, then 1 key to have a L1 symbol.)

3. Insert a new list (optional): STAT �EDIT, use the up arrow to place the cursor on a list name, then press INS [ 2nd DEL ] . Type the name of a list using the alpha character keys. The ALPHA key is locked down for you. Press ENTER. The new list is placed just before the point where the cursor was. To obtain a quick statistics, just use one of the build-in list L1 through L6 to enter the data, you do not need to create a new list with a name.

Obtain Numeric Measures from TI

Calculator

1. After entering data, return to home screen by pressing QUIT[2nd MODE].

2. Press STAT Key, select CALC menu, then choose the number 1 operation : 1-Var Stats, then ENTER . Enter the name of the list, say L1. That is,

STAT� CALC �1� ENTER � L1

Note: L1 is the default list. You do not need to enter it, if the data is on L1

Obtain Statistics from a Frequency

Distribution

• Enter the values in one list, say L1, and their

corresponding frequencies in another list, say L2.

Then,

STAT� CALC �1� ENTER � L1, L2

Note: Need to enter comma L2 after L1. The calculator will use the second list as the frequency for the values

entered on its list before to calculate the appropriate

statistics.

20

Example 1

63 64 76 76 81 83 85 86 88 89 90 91 92 93 93 93

94 97 99 99 99 101 108 109 112

Example: A random sample of students in a sixth grade class was selected.Their weights are given in the table below. Find the mean and variance, standard deviation, 5-number summary for this data using the TI calculator:

The output shows:

112max

99

92

84

63min

25

...996.11

...244.12

208083

2261

44.90

3

1

2

=

=

=

=

=

=

=

=

=

=

=

∑∑

X

Q

Med

Q

X

n

S

x

x

x

x

x

σ

Note:

1. Since this a sample data, we take Sx as

the standard deviation.

2. You may need to press the arrow key

on the calculator several times to view

these many statistics.

Example 2

7531Midpoint

1673Frequency

6 – 7.94 – 5.92 – 3.90 – 1.9Class

7max

5

3

3

1min

17

..647.1

..697.1

265

61

..588.3

3

1

2

=

=

=

=

=

=

=

=

=

=

=

∑∑

X

Q

Med

Q

X

n

S

x

x

x

x

x

σ

Consider the grouped data we considered previously:

Use TI calculator to obtain the statistics:

The output shows:Note: Here, the notations used in the

calculator correspond to the notations

used in the formula for computing mean,

variance and standard deviation of a

frequency distribution:

fxx

fxx

fn

∑∑∑ ∑∑

=

=

=

22