An article on peanut butter reported the following scores (quality ratings on a scale of 0 to 100) f...
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An article on peanut butter reported the following scores (quality ratings on a scale of 0 to 100) for various brands. Construct a comparative stem-and-leaf plot and compare the graphs. Creamy:56446236395350654540566841304050563022

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Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22

An article on peanut butter reported the following scores (quality ratings on a scale of 0 to 100) for various brands. Construct a comparative stem-and-leaf plot and compare the graphs.

  • Creamy:56446236395350654540566841304050563022

  • Crunchy:625375424740346252503442367580475662


Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22

  • Creamy:56446236395350654540566841304050563022

  • Crunchy:625375424740346252503442367580475662

Center:The center of the creamy is roughly 45 whereas the center for crunchy is higher at 51.

Shape: Both are unimodal but crunchy is skewed to the right while creamy is more symmetric.

Spread: The range for creamy and crunchy are equal at. There doesn’t seem to be any gaps in the distribution.


Variation

Variation


Which brand of paint is better why

Which Brand of Paint is better? Why?


Standard deviation

Standard Deviation

  • It’s a measure of the typical or average deviation (difference) from the mean.


Variance

Variance

  • This is the average of the squared distance from the mean.


Which brand of paint is better why1

Which Brand of Paint is better? Why?


Does the average help

Does the Average Help?

  • Paint A: Avg = 210/6 = 35 months

  • Paint B:Avg = 210/6 = 35 months

  • They both last 35 months before fading. No help in deciding which to buy.


Consider the spread

Consider the Spread

  • Paint A:Spread = 60 – 10 = 50 months

  • Paint B:Spread = 45 – 25 = 20 months

  • Paint B has a smaller variancewhich means that it performs more consistently. Choose paint B.


Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22

Formula for Population

Variance =

Standard Deviation =


Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22

Formula for Sample

Variance =

Standard Deviation =


Formulas for variance and st deviation

Formulas for Variance and St. Deviation

Population

Sample

Variance

Variance

Standard Deviation

Standard Deviation


Standard deviation1

Standard Deviation

  • A more powerful approach to determining how much individual data values vary.

  • This is a measure of the average distance of the observations from their mean.

  • Like the mean, the standard deviation is appropriate only for symmetric data!

  • The use of squared deviations makes the standard deviation even more sensitive than the mean to outliers!


Standard deviation2

Standard Deviation

  • One way to think about spread is to examine how far each data value is from the mean.

  • This difference is called a deviation.

  • We could just average the deviations, but the positive and negative differences always cancel each other out! So, the average deviation is always 0  not very helpful!


Finding variance

Finding Variance

  • To keep them from canceling out, we squareeach deviation.

  • Squaring always gives a positive value, so the sum will not be zero!

  • Squaring also emphasizes larger differences – a feature that turns out to be good and bad.

  • When we add up these squared deviations and find their average (almost), we call the result the variance.


Finding standard deviation

Finding Standard Deviation

  • This is the average of the squareddistance from the mean.

  • Variance will play an important role later – but it has a problem as a measure of spread.

  • Whatever the units of the original data are, the variance is in squared units – we want measures of spread to have the same units as the data, so to get back to the original units, we take the square root of .

  • The result is, s, is the standard deviation.


Let s look at the data again on the number of pets owned by a group of 9 children

Let’s look at the data again on the number of pets owned by a group of 9 children.

Recall that the mean was 5 pets.

Let’s take a graphical look at the “deviations” from the mean:


Let s find the standard deviation and variance of the data set of pets

Let’s Find the Standard Deviation and Variance of the Data Set of Pets

1 – 5 = -4

3 – 5 = -2

4 – 5 = -1

4 – 5 = -1

4 – 5 = -1

5 – 5 = 0

7 – 5 = 2

8 – 5 = 3

9 – 5 = 4


Find variance

Find Variance:

This is the “average” squared deviation.


Find the standard deviation

Find the Standard Deviation:

This 2.55 is roughly the average distance of the values in the data set from the mean.


Find the standard deviation and variance

Find the Standard Deviation and Variance


Homework

Homework

  • Worksheet


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