Review

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# Review - PowerPoint PPT Presentation

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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Presentation Transcript
Review
• Measures of central tendency
• Mean
• Median
• Mode
• Measures of dispersion or variation
• Range
• Variance
• 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 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)

Relative Frequency

0.5

0.4

0.3

0.2

0.1

1

2

3

4

5

0

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 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

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

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

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-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

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

Another method for detecting outliers involves percentiles.

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

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

Memorize:

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

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

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 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

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 Percentiles

The interquartile range (IQR) is defined to be:

IQR = QU -QL

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 - 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 - 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 - 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 - 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 - 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 - 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.

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.

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

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

• MIN, QL , M, QU, Max
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 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 - 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 - 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 - 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 - 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

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
The content structure of the provided text passage is a PowerPoint presentation on statistical concepts, including measures of central tendency, measures of dispersion, Chebyshev’s Theorem, the Empirical Rule, Z-scores, percentiles, and boxplots. The passage includes several examples and definitions.
multiple_questions_with_answers

the passage contains multiple mathematical questions and their answers, such as:
- calculating z-scores for specific data points.
- finding quartiles and the interquartile range (iqr) for a given dataset.
- identifying suspected and highly suspected outliers using the iqr method.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Review

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# Review - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Review
• Measures of central tendency
• Mean
• Median
• Mode
• Measures of dispersion or variation
• Range
• Variance
• 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 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)

Relative Frequency

0.5

0.4

0.3

0.2

0.1

1

2

3

4

5

0

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 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

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

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

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-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

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

Another method for detecting outliers involves percentiles.

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

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

Memorize:

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

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

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 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

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 Percentiles

The interquartile range (IQR) is defined to be:

IQR = QU -QL

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 - 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 - 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 - 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 - 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 - 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 - 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.

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.

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

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

• MIN, QL , M, QU, Max
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 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 - 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 - 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 - 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 - 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

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
user
The content structure of the provided text passage is a PowerPoint presentation on statistical concepts, including measures of central tendency, measures of dispersion, Chebyshev’s Theorem, the Empirical Rule, Z-scores, percentiles, and boxplots. The passage includes several examples and definitions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

The passage contains multiple mathematical questions and their answers, such as:
- Calculating z-scores for specific data points.
- Finding quartiles and the interquartile range (IQR) for a given dataset.
- Identifying suspected and highly suspected outliers using the IQR method.