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: Crunchy:

Creamy: Crunchy: 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 46. There doesn’t seem to be any gaps in the distribution.

Variation

Objective Calculate standard deviation and variance.

Relevance To be able to analyze a set of data and see how far each value is above or below the mean of the set.

Which Brand of Paint is better? Why? Brand A Brand B

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

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

Formula for Population Variance = Standard Deviation =

Formula for Sample Variance = Standard Deviation =

Formulas for Variance and St. Deviation Population Sample Variance Standard Deviation Variance Standard Deviation

Standard Deviation Facts 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! (Affected by extreme values)

Standard Deviation Facts 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!

Variance Facts To keep them from canceling out, we square each deviation. Squaring always gives a positive value, so the sum will not be zero! Affected by extreme values. When we add up these squared deviations and find their average (almost), we call the result the variance.

Variance

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 Pets x Sum = 52 1 – 5 = -4 3 – 5 = -2 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 ValuesDeviationsSquared Deviations Mean = Sum = 80

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