An article on peanut butter reported the following

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

Creamy: 56 65 40 44 45 50 62 40 56 36 56 30 39 68 22 53 41 50 30 Crunchy: 62 62 80 53 52 47 75 50 56 42 34 62 47 42 40 36 34 75 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? Brand A Brand B 10 35 60 45 50 30 30 35 40 40 20 25

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? Brand A Brand B 10 35 60 45 50 30 30 35 40 40 20 25

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 variance which 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 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 square each 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

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 x 1 1 – 5 = -4 3 3 – 5 = -2 4 4 – 5 = -1 5 5– 5=0 7– 5 =2 7 8 8– 5=3 9 9– 5=4 Sum = 16