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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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: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


Which Brand of Paint is better? Why?


Standard Deviation

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


Variance

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


Which Brand of Paint is better? Why?


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

  • 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.


Formula for Population

Variance =

Standard Deviation =


Formula for Sample

Variance =

Standard Deviation =


Formulas for Variance and St. Deviation

Population

Sample

Variance

Variance

Standard Deviation

Standard Deviation


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 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

  • 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

  • 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.

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

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:

This is the “average” squared 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


Homework

  • Worksheet


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