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

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Presentation Transcript
slide1

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: 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 56 30 22
  • Crunchy: 62 53 75 42 47 40 34 62 52 50 34 42 36 75 80 47 56 62
slide2

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

  • Crunchy: 62 53 75 42 47 40 34 62 52 50 34 42 36 75 80 47 56 62

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.

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.
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.
slide10

Formula for Population

Variance =

Standard Deviation =

slide11

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.

homework
Homework
  • Worksheet