1.2 Measuring Spread
AP Statistics
1.2 Measuring Spread
Variance and Standard Deviation
Objectives: Given a data set, compute the standard deviation and variance as measures of spread.
Give two reasons why we use squared deviations rather than just average deviations from the mean.
Explain what is meant by degrees of freedom.
Identify situations in which the standard deviation is the most appropriate measure of spread and situations in which the IQR is the most appropriate measure.
Standard Deviation (s):
The five-number summary is not the most common numerical description of a distribution. That belongs to the combination of the mean to measure center and the standard deviation to measure spread. The standard deviation measures spread by looking at:
How far the observations are from their mean.
Some deviations will be positive and some negative because some of the observations fall on each side of the mean. In fact the sum of the deviations of the observations from their mean will always be zero.
Squaring the deviations makes them all positive.
The Variance is the average squared deviation, therefore s2 and s will be large if the observations are widely spread about their mean, and small if the observations are all close to the mean.
Example 1:
Radon is a naturally occurring gas and is the second leading cause of lung cancer in the United States. It comes from the natural breakdown of uranium in the soil and enters buildings through cracks and other holes in the foundations. Found throughout the United States, levels vary considerably from state to state. There are several methods to reduce the levels of radon in your home, and the Environmental Protection Agency recommends using one of these if the measured level in your home is above 4 picocuries per liter. Four readings from Franklin County, Ohio, where the county average is 9.32 picocuries per liter, were 5.2, 13.8, 8.6 and 16.8.
a)Find the mean step-by-step.
b) Find the standard deviation step-by-step.
Finally, let’s find the standard deviation s:
c) Now enter the data into your calculator and use the mean and standard deviation buttons to obtain x̅ and s. Do the results agree with your hand calculations?
STAT Edit enter your data into L3 STAT CALC 1-Var Stats See your mean and Standard Deviation.
Degrees of Freedom: the number n – 1 is called the degrees of freedom of the variance or standard deviation. Many calculators offer a choice between dividing by n and dividing by n – 1, so be sure to use n – 1.
The use of squared deviations renders s even more sensitive than x̅ to a few extreme observations.
Choosing a Summary:
The five-number summary is usually better than the mean and standard deviation for describing a skewed distribution or a distribution with strong outliers. Use x̅ and s only for reasonably symmetric distributions that are free of outliers.
Example for choosing summaries:
Stemplots of annual returns for stocks and Treasury bills, 1950 to 2003. (a) Stock returns, in whole percents. (b) Treasury bill returns, in percents and tenths of percent. Which one would you use s and x̅, and which one would you use the five-number summary?
You see that returns on Treasury bills have a right skewed distribution. Convention in the financial world calls forx̅ and s because some parts of investment theory use them. For describing this right-skewed distribution, however, the five-number summary would be more informative.
Remember! The choice of using mean-based (mean, standard deviation) or median-based (median, five-number summary, IQR) statistics to describe a distribution should be based on the SHAPE of the distribution!
HW: pg. 89-90; 1.39 – 1.44