Download 5.7 Describing Variability: The Standard Deviation Although the

5.7 Describing Variability: The Standard Deviation
Although the five­number summary is the most generally useful numerical description of a distribution, it is not the most common. The most common combination of center and variability is the mean (μ)and standard deviation (σ). The standard deviation measures variability by looking at how far the observations are from their mean.
To calculate the standard deviation, we do the following steps:
1) Calculate the mean (sum of all observations divided by n, where n is the number of observations).
2) Find the deviation from the mean for each value, which is "observation minus mean". 3) Square each of these deviations, which will give us all positive numbers. 4) Add all of these positive numbers together.
5) Divide the sum by (n­1), where n is the number of observations.
6) Take the square root of this number.
Example 1: Find the mean (μ) and standard deviation (σ) for the following data set.
{10, 30, 50, 70}
n = 1
Example 2: The table below shows the total returns in % for a mutual fund for the 7­year period from 2003­2009. Find the mean and standard deviation for the returns.
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More important than the details of hand calculation are the properties that determine the usefulness of the standard deviation.
• The standard deviation (σ) measures variability about the mean μ. Use σ to describe the variability of a distribution only when you use μ to describe the center.
• σ = 0 only when there is no variability. This only happens when every observation is equal to the mean. Otherwise, σ > 0. As the observations display more variability about their mean, σ gets larger.
• σ has the same units of measurement as the original observations. For example, if you measure height in inches, then μ and σ are both in inches.
The use of squared deviations makes σ even more sensitive than μ to a few extreme •observations. Distributions with outliers and strongly skewed distributions have large standard deviations. The number σ does not give much helpful information in these cases. Choosing a summary:
The five­number summary is usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use μ and σ only for reasonably symmetric distributions that are free of outliers.
Remember that a graph gives the best overall picture of a distribution. Numerical measures of center and variability report specific facts about a distribution , but they do not describe its entire shape; for example, numerical summaries do not disclose the presence of clusters. Thus, the best place to begin is with a graph of the data!
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Example 3: Calculate the mean and standard deviation of the ages of a group of students as shown below, and give the units for each.
{19, 18, 22, 47, 20, 18, 18, 18, 19, 18, 18, 19}
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Example 4: Calculate the mean and standard deviation for the number of firearms per 1000 people in the 10 states listed below, and give the units for each.
Homework: pp. 203-204: 34 and 38 plus worksheet
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