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Review. Measures of central tendency Mean Median Mode Measures of dispersion or variation Range Variance Standard Deviation. Interpreting the Standard Deviation. Chebyshev’s Theorem

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Review

Review

  • Measures of central tendency

    • Mean

    • Median

    • Mode

  • Measures of dispersion or variation

    • Range

    • Variance

    • Standard Deviation


Interpreting the standard deviation

Interpreting the Standard Deviation

Chebyshev’s Theorem

The proportion (or fraction) of any data set lying within K standard deviations of the mean is always at least 1-1/K2, where K is any positive number greater than 1.

For K=2 we obtain, at least 3/4 (75 %) of all scores will fall within 2 standard deviations of the mean, i.e. 75% of the data will fall between


Interpreting the standard deviation1

Interpreting the Standard Deviation

Chebyshev’s Theorem

The proportion (or fraction) of any data set lying within K standard deviations of the mean is always at least 1-1/K2, where K is any positive number greater than 1.

For K=3 we obtain, at least 8/9 (89 %) of all scores will fall within 3 standard deviations of the mean, i.e. 89% of the data will fall between


This data is symmetric bell shaped or normal data

This Data is Symmetric, Bell Shaped (or Normal Data)

Relative Frequency

0.5

0.4

0.3

0.2

0.1

1

2

3

4

5

0


The empirical rule

The Empirical Rule

The Empirical Rule states that for bell shaped (normal) data:

68% of all data points are within 1 standard deviations of the mean

95% of all data points are within 2 standard deviations of the mean

99.7% of all data points are within 3 standard deviations of the mean


The empirical rule1

The Empirical Rule

The Empirical Rule states that for bell shaped (normal) data, approximately:

68% of all data points are within 1 standard deviations of the mean

95% of all data points are within 2 standard deviations of the mean

99.7% of all data points are within 3 standard deviations of the mean


Z score

Z-Score

To calculate the number of standard deviations a particular point is away from the standard deviation we use the following formula.


Z score1

Z-Score

To calculate the number of standard deviations a particular point is away from the standard deviation we use the following formula.

The number we calculate is called the z-score of the measurement x.


Example z score

Example – Z-score

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.

a) Find the z-score of the data point 35.

b) Find the z-score of the data point 73.


Example z score1

Example – Z-score

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.

  • Find the z-score of the data point 35.

    z = -2.11

    b) Find the z-score of the data point 73.

    z = 0.16


Interpreting z scores

Interpreting Z-scores

The further away the z-score is from zero the more exceptional the original score.

Values of z less than -2 or greater than +2 can be considered exceptional or unusual (“a suspected outlier”).

Values of z less than -3 or greater than +3 are often exceptional or unusual (“a highly suspected outlier”).


Percentiles

Percentiles

Another method for detecting outliers involves percentiles.


Percentiles1

Percentiles

Another method for detecting outliers involves percentiles.

The pth percentile ranking is a number so that p% of the measurements fall below the pth percentile and 100 – p% fall above it.


Example

Example

If your score on a class quiz of 200 students places you in the 80th percentile, then only 40 students received a higher mark then you


Important percentiles

Important Percentiles

Memorize:

The 25th percentile is called the lower quartile (QL)

The 75th percentile is called the upper quartile (QU)


Important percentiles1

Important Percentiles

Memorize:

The 25th percentile is called the lower quartile (QL)

The 75th percentile is called the upper quartile (QU)

The 50th percentile is called the


Important percentiles2

Important Percentiles

Memorize:

The 25th percentile is called the lower quartile (QL)

The 75th percentile is called the upper quartile (QU)

The 50th percentile is called the median (M)


Quick way to find quartiles

Quick way to find quartiles

Arrange the data in increasing order

The middle number (or average of the two middle numbers) is the 50th percentile.

Find the middle number in the set of numbers greater than the median. This is the upper quartile.

Similarly, find the lower quartile


Important percentiles3

Important Percentiles

The interquartile range (IQR) is defined to be:

IQR = QU -QL


Example fax

Example - Fax


Example fax1

Example - Fax

Here are the number of pages faxed by each fax sent from our Math and Stats department since April 24th, in the order that they occurred.

5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25


Example fax2

Example - Fax

Here are the number of pages faxed by each fax sent from our Math and Stats department since April 24th, in the order that they occurred.

5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25

Find QU , QL , M and IQR.


Example fax3

Example - Fax

1) Rank the n points of data from lowest to highest

5, 1, 2, 6, 10, 3, 6, 2, 2, 2, 2, 2, 2, 4, 5, 1, 13, 2, 5, 5, 1, 3, 6, 37, 2, 8, 2, 25


Example fax4

Example - Fax

1) Rank the n points of data from lowest to highest

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37


Example fax5

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

To compute QU and QL , M.

Find the Median, divide the data into two equal parts and then the Medians of these.


Example fax6

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

N = 28

Therefore, median is half way between the 14th and 15th number.

Median = 50th percentile = 3


Example fax7

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3|3, 4, 5, 5

5, 5, 6, 6, 6, 8, 10, 13, 25, 37

M = 3

QU = 6

QL = 2

IQR=6-2=4.


Percentiles2

Percentiles

Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not effected at all by outliers.


Percentiles3

Percentiles

Sometimes the IQR, is a better measure of variance then the standard deviation since it only depends on the center 50% of the data. That is, it is not effected at all by outliers.

To use the IQR as a measure of variance we need to find the Five Number Summary of the data and then construct a Box Plot.


Five number summary and outliers

Five Number Summary and Outliers

The Five Number Summary of a data set consists of five numbers,

  • MIN, QL , M, QU, Max


Five number summary and outliers1

Five Number Summary and Outliers

The Five Number Summary of a data set consists of five numbers,

  • MIN, QL , M, QU, Max

    Suspected Outliers lie

  • Above 1.5 IQRs but below 3 IQRs from the Upper Quartile

  • Below 1.5 IQRs but above 3 IQRs from the Lower Quartile

    Highly Suspected Outliers lie

  • Above 3 IQRs from the Upper Quartile

  • Below 3 IQRs from the Lower Quartile.


Five number summary and outliers2

Five Number Summary and Outliers

The Inner Fences are:

  • data between the Upper Quartile and 1.5 IQRs above the Upper Quartile and

  • data between the Lower Quartile and 1.5 IQRs below the Lower Quartile

    The Outer Fences are:

  • data between 1.5 IQRs above the Upper Quartile and 3 IQRs above the Upper Quartile and

  • data between 1.5 IQRs Lower Quartile and 3 IQRs below the Lower Quartile


Example fax8

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,

3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

Min=1, QL = 2, M = 3, QU = 6, Max = 37.

IQR=6-2=4.


Example fax9

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,

3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

Min=1, QL = 2, M = 3, QU = 6, Max = 37.

IQR=6-2=4.

Inner Fence extremes: -4, 12


Example fax10

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,

3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

Min=1, QL = 2, M = 3, QU = 6, Max = 37.

IQR=6-2=4.

Inner Fence extremes: -4, 12

Outer Fence extremes: -10, 18


Example fax11

Example - Fax

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,

3, 4, 5, 5, 5, 5, 6, 6, 6, 8, 10, 13, 25, 37

Min=1, QL = 2, M = 3, QU = 6, Max = 37.

IQR=6-2=4.

Inner Fence extremes: -4, 12

Outer Fence extremes: -8, 18

Suspected Outliers: 13

Highly Suspected Outliers: 25, 37


Definition boxplot

Definition: Boxplot

A boxplotis a graph of lines (from lowest point inside the lower inner fence to highest point in the upper inner fence) and boxes (from Lower Quartile to Upper quartile) indicating the position of the median.

*

Lowest data

Point more than

the lower inner

fence

Highest data

Point less than

the upper inner

fence

Median

Upper

Quartile

Lower

Quartile

Outliers


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