1

Review• Descriptive Statistics

– Qualitative (Graphical)– Quantitative (Graphical)– Summation Notation– Qualitative (Numerical)

• Central Measures (mean, median, mode and modal class)

• Shape of the Data

• Measures of Variability

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Outlier

A data measurement which is unusually large or small compared to the rest of the data.

Usually from:– Measurement or recording error– Measurement from a different population– A rare, chance event.

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Advantages/Disadvantages Mean

• Disadvantages– is sensitive to outliers

• Advantages– always exists– very common– nice mathematical properties

4

Advantages/Disadvantages Median

• Disadvantages– does not take all data into account

• Advantages– always exists– easily calculated– not affected by outliers– nice mathematical properties

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Advantages/Disadvantages Mode

• Disadvantages– does not always exist, there could be just one

of each data point– sometimes more than one

• Advantages– appropriate for qualitative data

6

Review

A data set is skewed if one tail of the distribution has more extreme observations than the other.

http://www.shodor.org/interactivate/activities/SkewDistribution/

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Review

Skewed to the right: The mean is bigger than the median.

xM

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Review

Skewed to the left: The mean is less than the median.

x M

9

Review

When the mean and median are equal, the data is symmetric

Mx

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Numerical Measures of Variability

These measure the variability or spread of the data.

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Numerical Measures of Variability

These measure the variability or spread of the data.

Relative Frequency

0 1 3 4 52

0.3

0.4

0.5

0.2

0.1

Mx

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Numerical Measures of Variability

These measure the variability or spread of the data.

Relative Frequency

0 1 3 4 52

0.3

0.4

0.5

0.2

0.1

Mx

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Numerical Measures of Variability

These measure the variability or spread of the data.

Relative Frequency

0 1 3 4 52

0.3

0.4

0.5

0.2

0.1

6 7

Mx

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Numerical Measures of Variability

These measure the variability, spread or relative standing of the data.

– Range– Standard Deviation– Percentile Ranking– Z-score

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Range

The range of quantitative data is denoted R and is given by:

R = Maximum – Minimum

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Range

The range of quantitative data is denoted R and is given by:

R = Maximum – Minimum

In the previous examples the first two graphs have a range of 5 and the third has a range of 7.

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Range

R = Maximum – Minimum

Disadvantages: – Since the range uses only two values in the

sample it is very sensitive to outliers.– Give you no idea about how much data is in the

center of the data.

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What else?

We want a measure which shows how far away most of the data points are from the mean.

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What else?

We want a measure which shows how far away most of the data points are from the mean.

One option is to keep track of the average distance each point is from the mean.

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Mean Deviation

The Mean Deviation is a measure of dispersion which calculates the distance between each data point and the mean, and then finds the average of these distances.

n

xx

n

xx ii

sumDeviation Mean

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Mean Deviation

Advantages: The mean deviation takes into account all values in the sample.

Disadvantages: The absolute value signs are very cumbersome in mathematical equations.

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Standard Deviation

The sample variance, denoted by s², is:

1

)( s

22

n

xxi

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Standard Deviation

The sample variance, denoted by s², is:

The sample standard deviation is

The sample standard deviation is much more commonly used as a measure of variance.

.2ss

1

)( s

22

n

xxi

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Example

Let the following be data from a sample:

2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

Find:

a) The range

b) The standard deviation of this sample.

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Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

a) The range

b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2

x

R

ix

)( xxi 2)( xxi

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Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. a) The range

b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2

310

30

10

2541252342

x

415R

ix

)( xxi 2)( xxi

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Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. a) The range

b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2

-1 1 0

310

30

10

2541252342

x

415R

ix

)( xxi 2)( xxi

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Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. a) The range

b) The standard deviation of this sample.

2 4 3 2 5 2 1 4 5 2

-1 1 0 -1 2 -1 -2 1 2 -1

1 1 0 1 4 1 4 1 4 1

310

30

10

2541252342

x

415R

ix

)( xxi 2)( xxi

29

Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. 2 4 3 2 5 2 1 4 5 2

-1 1 0 -1 2 -1 -2 1 2 -1

1 1 0 1 4 1 4 1 4 1

ix

)( xxi 2)( xxi

1

)( s

22

n

xxi

30

Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. 2 4 3 2 5 2 1 4 5 2

-1 1 0 -1 2 -1 -2 1 2 -1

1 1 0 1 4 1 4 1 4 1

ix

)( xxi 2)( xxi

110

1414141011

1

)( s

22

n

xxi

31

Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2. 2 4 3 2 5 2 1 4 5 2

-1 1 0 -1 2 -1 -2 1 2 -1

1 1 0 1 4 1 4 1 4 1

ix

)( xxi 2)( xxi

2110

1414141011

1

)( s

22

n

xxi

32

Sample: 2, 4, 3, 2, 5, 2, 1, 4, 5, 2.

2110

1414141011

1

)( s

22

n

xxi

41.12 ss 2

Standard Deviation:

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More Standard DeviationLike the mean, we are also interested in the population variance (i.e. your sample is the whole population) and the population standard deviation.

The population variance and standard deviation are denoted σ and σ2 respectively.

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More Standard DeviationThe population variance and standard deviation are denoted σ and σ2 respectively.

****The formula for population variance is slightly different than sample variance

nn

xx

n

xxi

ii

2

22

2 )(

2

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Example – Using Standard Deviation

35, 59, 70, 73, 75, 81, 84, 86.

The mean and standard deviation are 70.4 and 16.7, respectively.

We wish to know if any of are data points are outliers. That is whether they don’t fit with the general trend of the rest of the data.

To find this we calculate the number of standard deviations each point is from the mean.

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Example – Using Standard Deviation

To find this we calculate the number of standard deviations each point is from the mean.

To simplify things for now, work out which data points are within

a) one standard deviation from the mean i.e.

b) two standard deviations from the mean i.e.

c) three standard deviations from the mean i.e.

) ,( sxsx

)2 ,2( sxsx

)3 ,3( sxsx

37

Example – Using Standard Deviation

Here are eight test scores from a previous Stats 201 class:

35, 59, 70, 73, 75, 81, 84, 86.

The mean and standard deviation are 70.4 and 16.7, respectively. Work out which data points are within

a) one standard deviation from the mean i.e.

b) two standard deviations from the mean i.e.

c) three standard deviations from the mean i.e.

)1.87 ,7.53()7.160.47 ,7.164.70(

)8.301 ,0.37())7.16(20.47 ),7.16(24.70(

)5.021 ,3.21())7.16(30.47 ),7.16(34.70(

38

Example – Using Standard Deviation

Here are eight test scores from a previous Stats 201 class:

35, 59, 70, 73, 75, 81, 84, 86.

The mean and standard deviation are 70.4 and 16.7, respectively. Work out which data points are within

a) one standard deviation from the mean i.e.

59, 70, 73, 75, 81, 84, 86

b) two standard deviations from the mean i.e.

59, 70, 73, 75, 81, 84, 86

c) three standard deviations from the mean i.e.

35, 59, 70, 73, 75, 81, 84, 86