Chap03

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CHAPTER 3

Data Description

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Objectives

Summarize data using measures of centraltendency, such as the mean, median, mode,

and midrange.

Describe data using the measures of variation, such as the range, variance, and

standard deviation.

Identify the position of a data value in a dataset using various measures of position, such

as percentiles, deciles, and quartiles.

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Objectives (cont’d.)

Use the techniques of exploratory dataanalysis, including stem-and-leaf plots,

boxplots, and five number summaries to

discover various aspects of data.

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Introduction

Statistical methods can be used to summarizedata.

Measures of averages are also calledmeasures

of central tendency and include themean,

median,mode, andmidrange.

Measures that determine the spread of data

values are calledmeasures of variation or

measures of dispersion and include therange,

variance, andstandard deviation.

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Introduction (cont’d.)

Measures of position tell where a specific data value falls within the data set or its relative

position in comparison with other data

values.

The most common measures of position are

percentiles,deciles, andquartiles.

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The measures of central tendency, variation,and position are part of what is called

traditional statistics. This type of data is

typically used to confirm conjectures about

the data.

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Another type of statistics is calledexploratorydata analysis. These techniques include the

stem-and-leaf plot, theboxplot, and the five-

number summary. They can be used to exploredata to see what they show.

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asic !ocabu"ar#

Astatistic is a characteristic or measureobtained by using the data values from a

sample.

A parameter is a characteristic or measureobtained by using all the data values for a

specific population.

When the data in a data set is ordered it is

called adata array.

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$enera" Roundin% Ru"e

In statistics the basicrounding rule is that

when computations

are done in thecalculation, rounding

should not be done

until the final answeris calculated.

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Centra" Tendenc#& T'e ean

Themean, also known as thearithmeticaverage, is the sum of the values divided by

the total number of values. Th

Rounding rule: the mean should be rounded

to one more decimal place than occurs in the

raw data.

The type of mean that considers an additional

factor is called theweighted mean.

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Centra" Tendenc#& T'e ean (cont’d.)

One computes the mean by using all the values of the data.

The mean varies less than the median or

mode when samples are taken from the same

population and all three measures are

computed for these samples.

The mean is used in computing other

statistics, such as variance.

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Centra" Tendenc#& T'e ean (cont’d.)

The mean for the data set is unique, and notnecessarily one of the data values.

The mean cannot be computed for an open-

ended frequency distribution.

The mean is affected by extremely high or low

values and may not be the appropriateaverage to use in these situations.

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Centra" Tendenc#& edian and ode

Themedian is the halfway point in a data set. The symbol for the median is MD.

The median is found by arranging the data in

order and selecting the middle point.

The value that occurs most often in a data set

is called themode.

The mode for grouped data, or the class with

the highest frequency, is the modal class.

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Centra" Tendenc#& T'e edian

The median is used when one must find thecenter of middle value of a data set.

The median is used when one must determine

whether the data values fall into the upper half orlower half of the distribution.

The median is used to find the average of an open-

ended distribution. The median is affected less than the mean by

extremely high or extremely low values.

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Centra" Tendenc#& T'e ode

The mode is used when the most typical caseis desired.

The mode is the easiest average to compute.

The mode can be used when the data are

nominal, such as religious preference, gender,

or political affiliation.

The mode is not always unique. A data set can

have more than one mode, or the mode may

not exist for a data set.

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Centra" Tendenc#& T'e idran%e

Themidrange is defined as the sum of thelowest and highest values in the data set

divided by 2.

The symbol for midrange is MR.

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Centra" Tendenc#& T'e idran%e (cont’d.)

The midrange is easy to compute.

The midrange gives the midpoint.

The midrange is affected by extremely high orlow values in a data set.

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Distribution 'apes

In a positively skewedor right skeweddistribution, the majority of the data values

fall to the left of the mean and cluster at the

lower end of the distribution.

i h h ill i i i i d

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Distribution 'apes (cont’d.)

In asymmetrical distribution, the data valuesare evenly distributed on both sides of the

mean.

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Distribution 'apes (cont’d.)

When the majority of the data values fall tothe right of the mean and cluster at the upper

end of the distribution, with the tail to the

left, the distribution is said to benegativelyskewed orleft skewed.

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T'e Ran%e

Therange is the highest value minus thelowest value in a data set.

The symbolR is used for the range.

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!ariance and tandard Deviation

Thevariance is the average of the squares ofthe distance each value is from the mean. The

symbol for the population variance isσ 2.

Thestandard deviation is the square root of

the variance. The symbol for the population

standard deviation isσ .Rounding rule: The

final answer should be rounded to one more

decimal place than the original data.

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Coe**icient o* !ariation

Thecoefficient of variation is the standarddeviation divided by the mean. The result is

expressed as a percentage.

The coefficient of variation is used to compare

standard deviations when the units are

different for the two variables being compared.


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!ariance and tandard Deviation

Variances and standard deviations can be

used to determine the spread of the data. If

the variance or standard deviation is large,

the data are more dispersed. The information

is useful in comparing two or more data sets

to determine which is more variable.

The measures of variance and standarddeviation are used to determine the

consistency of a variable.


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!ariance and tandard Deviation (cont’d.)

The variance and standard deviation can beused to estimate the percentage of data values

that fall within a specified interval in a

distribution.

The variance and standard deviation are used

quite often in inferential statistics.


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C'eb#s'ev’s T'eore+

The proportion of values from a data set that will fall withinkstandard deviations of the

mean will be at least 1 – 1/k2; wherek is a

number greater than 1.

This theorem applies to any distribution

regardless of its shape.


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E+pirica" Ru"e *or ,or+a" Distributions

Approximately 68% of the data values fall within one standard deviation of the mean.

Approximately 95% of the data values will fall

within two standard deviations of the mean.

Approximately 99.7% of the data values will

fall within three standard deviations of themean.

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tandard cores

Astandard score orz score is used whendirect comparison of raw scores is impossible.

A standard score orz score for a value is

obtained by subtracting the mean from the

value and dividing the result by the standard

deviation.


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Percenti"es

Percentiles are position measures used ineducational and health-related fields to

indicate the position of an individual in a

group.

Percentiles divide the data set into 100 equal

parts.

ThePth percentile is a value whereP% of the

data values are less than or equal to the value.


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-uarti"es and Deci"es

Quartiles divide the distribution into fourgroups. The quartiles are denoted by

Q1,Q2, andQ3. Note thatQ1 is the same

as the 25th percentile;Q2 is the same asthe 50th percentile or the median; and

Q3 corresponds to the 75th percentile.

Deciles divide the distribution into 10 groups.

They are denoted byD1,D2, etc.


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Out"iers

Anoutlier is an extremely high or an extremelylow data value when compared with the rest of

the data values.

Outliers can be the result of measurement orobservational error.

When a distribution is normal or bell-shaped,

data values that are beyond three standard

deviations of the mean can be considered

suspected outliers.


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Ep"orator# Data Ana"#sis

The purpose ofexploratory data analysis is toexamine data in order to find out what

information can be discovered. For example:

Are there any gaps in the data?

Can any patterns be discerned?


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te+/and/0ea* P"ots

Astem-and-leaf plot is a data plot that usespart of a data value as the stem and part of

the data value as the leaf to form groups or

classes.

It has the advantage over grouped frequency

distribution of retaining the actual data while

showing them in graphic form.


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op"ots and 1ive/,u+ber u++aries

Boxplots are graphical representations of a five-number summary of a data set. The five specific

values that make up a five-number summary are:

The lowest value of data set (minimum)

Q1(or 25th percentile)

The median (or 50th percentile)

Q3

(or 75th percentile) The highest value of data set (maximum)

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u++ar#

Some basic ways to summarize data includemeasures of central tendency, measures of

variation or dispersion, and measures of

position.

The three most commonly used measures of

central tendency are the mean, median, and

mode. The midrange is also used to represent

an average.


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u++ar# (cont’d.)

The three most commonly used measurementsof variation are the range, variance, and

standard deviation.

The most common measures of position are

percentiles, quartiles, and deciles.

Data values are distributed according toChebyshev’s theorem and in special cases, the

empirical rule.


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u++ar# (cont’d.)

The coefficient of variation is used to describethe standard deviation in relationship to the

mean.

These methods are commonly called traditionalstatistics.

Other methods, such as the stem-and-leaf plot,

the boxplot, and five-number summary, are part

of exploratory data analysis; they are used to

examine data to see what they reveal.


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Copyright © The McGraw Hill Companies, Inc. Permission required or

reproduction or display. Conc"usions

By combining all of thesetechniques together, the

student is now able to

collect, organize,summarize and present

data.

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