statistics class 4 february 11 th, 2012. group quiz 3 heights of statistics students were obtained...

Statistics Class 4

February 11th , 2012

Group Quiz 3

Heights of statistics students were obtained by the author as a part of  a study conducted for class.  The last digits of those heights are listed below.  Construct a frequency distribution and then a histogram (recommend 10 classes).  Based on the distribution, do the heights appear to be reported or actually measured?  What do you know about the accuracy of the results?

0 0 0 0 0 0 0 0 0 1 1 2 3 3 3 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 8 8 8 9

Measure of center

A measure of center is a value at the center or middle of a data set.

The mean of a set of data is the measure of center found by adding the data values and dividing the total by the number of data values.

Round 1 more decimal place than the data

Calculate the mean of some data

76 64 72 80 88 76 60 76 72 76 68 80 80 104 64 88 68 60 68 76

80 72 76 72 68 88 72 80 96 60 72 72 68 88 72 88 64 124 80 64

Calculate the mean of some data

76+64+72+80+88+76+60+76+72+76+68+80+80+104+64+88+68+60+68+76+80+72+76+72+68+88+72+80+96+60+72+72

+68+88+72+88+64+124+80+64 = 3052

x= 3052/40=76.3

Median

The median of a data set is the measure of center that is the middle value when the data values when arranged in increasing (decreasing) order.

So take 

76 64 72 80 88 76 60 76 72 76 68 80 80 104 64 88 68 60 68 76

80 72 76 72 68 88 72 80 96 60 72 72 68 88 72 88 64 124 80 64

order it

60 60 60 64 64 64 64 68 68 68 68 68 72 72 72 72 72 72 72 72 76 76 76 76 76 76 80 80 80 80 80 80 88 88 88 88 88 96 104 124  

Median

Select the middle values

60 60 60 64 64 64 64 68 68 68 68 68 72 72 72 72 72 72 72 72 76 76 76 76 76 76 80 80 80 80 80 80 88 88 88 88 88 96 104 124

Even number of data values average the middle two.

(72+76)/2=148/2=74

Mode

The mode of a data set is the data value that occurs with greatest frequency.

When two data values occur with the same greatest frequency, each one is a mode and the data set is bimodal.

No repeated data value, then there is no mode. Find the mode of our previous data

60 60 60 64 64 64 64 68 68 68 68 68 72 72 72 72 72 72 72 72 76 76 76 76 76 76 80 80 80 80 80 80 88 88 88 88 88 96 104 124 

Mode

The mode of a data set is the data value that occurs with greatest frequency.

When two data values occur with the same greatest frequency, each one is a mode and the data set is bimodal.

No repeated data value, then there is no mode. Find the mode of our previous data

60 60 60 64 64 64 64 68 68 68 68 68 72 72 72 72 72 72 72 72 76 76 76 76 76 76 80 80 80 80 80 80 88 88 88 88 88 96 104 124 

Mode is 72.

Midrange

The midrange of a data set is the value midway between the maximum and minimum values

midrange=(maximum data value + minimum data value)/2

From our previous data set the max was 124 and the min was 60 so....

midrange= (124 + 60)/2=62.

Weighted Mean

• Where, are the weights, and are the values.

Course Credits ( Grade (

English 3 C

Physics 4 B

Calculus 5 A

Chemistry 4 B

Weighted Mean

Course Credits ( Grade (

English 3 C

Physics 4 B

Calculus 5 A

Chemistry 4 B

What do you need on the final?

• Suppose you are in a class, where your grade depends on 4 tests each worth 100 points. Let’s say you earned a 95 on the first test, a 91 on the second test, and a 81 on the third test. What grade must you earn on the final test to receive an A (a mean test score of a 90)?

Mean from a frequency distribution

• is the estimate of the mean from grouped data “a.k.a a freq. Dist.”

• frequency of each class/bin

• midpoint of each class/bin• sum of the products (multiplication) of the each midpoint and

its frequency• total of the frequencies

Pulse Rate Freq

60-69 12

70-79 14

80-89 11

90-99 1

100-109 1

110-119 0

120-129 1

Total 40

Pulse Rate Freq Midpoint

60-69 12 64.5

70-79 14 74.5

80-89 11 84.5

90-99 1 94.5

100-109 1 104.5

110-119 0 114.5

120-129 1 124.5

Total 40

Pulse Rate Freq f Midpoint x   x*f

60-69 12 64.5 774

70-79 14 74.5 1043

80-89 11 84.5 929.5

90-99 1 94.5 94.5

100-109 1 104.5 104.5

110-119 0 114.5 0

120-129 1 124.5 124.5

Total 40

Pulse Rate Freq f Midpoint x   x*f

60-69 12 64.5 774

70-79 14 74.5 1043

80-89 11 84.5 929.5

90-99 1 94.5 94.5

100-109 1 104.5 104.5

110-119 0 114.5 0

120-129 1 124.5 124.5

Total 40 3070

Pulse Rate Freq f Midpoint x   x*f

60-69 12 64.5 774

70-79 14 74.5 1043

80-89 11 84.5 929.5

90-99 1 94.5 94.5

100-109 1 104.5 104.5

110-119 0 114.5 0

120-129 1 124.5 124.5

Totals 40 3070

Mean 3070/40 76.75

You try!

• Estimate the mean of the frequency Distribution Below

Tar(mg) in filtered cigarettes

Frequency

2-5 2

6-9 2

10-13 6

14-17 15

Homework

3-1: 1-21 odd

Measure of Variation

We continue our investigation of descriptive statistics.  This time we are going to study measure of variation.  These measures describe how our data is spread out.

Measure of Variation

The range of a set of data values is the difference between the maximum data value and the minimum data value.

range= (max data value) - (min data value)

Ex.  Find the the range of the values 3, 4, 6, 17.

Measure of Variation

The range of a set of data values is the difference between the maximum data value and the minimum data value.

range = (max data value) - (min data value)

Ex.  Find the the range of the values 3, 4, 6, 17.

range = (17) - (3) = 14

Measure of Variation

The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean.  It is a type of average derivation of values from the mean that is calculated by using the following formulas:

Measure of Variation

Shortcut formula:

 n = number of values (sample size)

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.

Step 1: Find the mean.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.

Step 1: Find the mean.

mean = (3+4+6+8+4)/5 = 25/5 = 5.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.

Step 1: Find the mean.

mean = (3+4+6+8+4)/5 = 25/5 = 5.

Step 2: Subtract mean from each sample value.

Measures of Variation

•

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4. 

Step 3: Square the deviations from step 2.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4. 

Step 3: Square the deviations from step 2.

(-2)2 = 4(-1)2 = 1(1)2 = 1(3)2 =9(-1)2 = 1

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 4: Sum the squares from step 3.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 4: Sum the squares from step 3.

 4 +1 + 1 + 9 + 1 = 16

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 4: Sum the squares from step 3.

 4 +1 + 1 + 9 + 1 = 16

Step 5: Divide by N-1.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 4: Sum the squares from step 3.

 4 +1 + 1 + 9 + 1 = 16

Step 5: Divide by N-1.

16/(5-1) = 16/4 = 4.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 6: Find the square root of 5.

Measures of Variation

Lets use the first formula to find the standard deviation of the following sample 3, 4, 6, 8, 4.  

Step 6: Find the square root of step 5.

Measures of variation

Your turn!

Find the variation of the following sample 2, 4, 5, 8, 7, 2, 3, 1.using both methods.

Measures of variation

Your turn!

Find the variation of the following sample 2, 4, 5, 8, 7, 2, 3, 1.using both methods.

s=2.5

Measures of Variation

The standard deviation of a population, is slightly different than that of a sample. Namely we divide by n instead of n-1.

population standard deviation 

Measures of Variation

The variance of a set of values is the measure of variance equal to the square of the standard deviation.

Measures of Variation

The variance of a set of values is the measure of variance equal to the square of the standard deviation.

The sample variance s2 is an unbiased estimator of the population variance.

Measures of Variation

The Range Rule of Thumb is a simple tool that allows you to either interpret the values of distribution if you know the standard deviation, or to estimate the standard deviation s if it is unknown.

Interpreting a Known Value of the Standard Deviation:

Estimating a Value of the Standard Deviation s:

+

Measures of Variation

The Wechsler Adult Intelligence Scale involves and IQ test designed so that the mean score is 100 and the standard deviation is 15. Use the range rule of thumb to find the minimum and maximum "usual" IQ scores.  Then determine whether an IQ score of 135 would be considered "unusual."

Measures of Variation

Use the range rule of thumb to estimate the standard deviation of the sample of 100 FICO credit rating scores listed in Data Set 24 in Appendix B. Those scores have a min of 444 and a max of 850.

Measures of Variation

Empirical Rule (68-95-99.7) for data with a Bell-Shaped Distribution.

• About 68% of all values fall within 1 standard deviations of the mean.

• About 95% of all values fall within 2 standard deviations of the mean.

• About 99.7% of all values fall within 3 standard deviations of the mean.

Measures of Variation

Empirical Rule (68-95-99.7) for data with a Bell-Shaped Distribution.

Measures of Variation

Empirical Rule (68-95-99.7) for data with a Bell-Shaped Distribution.

Measures of Variation

IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. What percentage of IQ scores are between 70 and 130?

Measures of Variation

IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. What percentage of IQ scores are between 70 and 130?

Coefficient of Variation

• The Coefficient of Variation or CV for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean, and is given by the following:

Sample

Population

Coefficient of Variation

Compare the variation in the heights of men to the variation in weights of men, using these sample results obtained from Data Set 1 in Appendix B: for men, the heights yield a mean of 68.34 in. and standard deviation 3.02 in; the weights yield a mean of 172.55 lbs. and a standard deviation of 26.33 lbs.

Coefficient of Variation

Compare the variation in the heights of men to the variation in weights of men, using these sample results obtained from Data Set 1 in Appendix B: for men, the heights yield a mean of 68.34 in. and standard deviation 3.02 in; the weights yield a mean of 172.55 lbs. and a standard deviation of 26.33 lbs.

heights: weights:

More Homework!

• 3-2: 7-15 odd, 29, 31, 41,