STATISTICS
       
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Measures of DeviationWhen you have described the sample, using the mean, mode or median, do you feel that there is more information that needs to be looked at if we want to compare the production of Screws and Bolts Inc. to the production of a different company? Is the mean, for example, always sufficient to determine which company has the best production?Take a look at the two samples in the table. They both have the same mean (to the nearest hundreth), but are the samples identical? How are they different?
The first thing we notice is that there is a wider range in the length of the bolts from Company 2. Some of their bolts are as short as 2.49 cm, and the longest are 2.56. The range of their lengths is 2.56-2.49=.07 cm. No bolts from Company 1 are shorter than 2.52, and their longest bolts are 2.56 cm. The range of their lengths is 2.56-2.52=.04 cm. Which do you think is better, to have a short range or a long range? Why? You will also notice, that although the range is longer for Company 2, there are few bolts whose lenght is far from the desired length. Does that make a difference? How could we describe the range, and also take into account how the lengths are distributed within the range? One way to do that is to calculate the standard deviation. The standard deviation is a neat statistic. Let's say that the mean length of a sample is 2.54, and the standard deviation is .03 cm. Then we know that 34.13% of the bolts in the sample have length between 2.54 and 2.54+.03=2.57. Another 34.13% of the bolts have lengths between 2.54 and 2.54-.03=2.51. In fact, we can determine how much of the sample lies in any part of the total range. We do this by checking how many standard deviations are between the mean and the endpoints of the interval we are interested in. The table in the reference sheet is used for that purpose. When we use the standard deviation in this way we are assuming that our data are similar to a hypothetical data set called the normal distribution. When you used your TI calculator to calculate the mean and the median of the lenghts of bolts in the sample from Screws and Bolts Inc., the standard deviation was calcuated at the same time. All you need to know is how to recognize it in the list of numbers your calculator gives you when you select CALCULATE 1-Var Stats. Just in case, here is a review of how to calculate 1-Var Stats.
 
The Z-Table
The table tells us the percentage of the data between
the mean and the number of standard deviations from the mean. For example,
if a number is .90 standard deviations from the mean, then 31.59% of the
data are between the mean and that number. If a number is 2.00 standard
deviations from the mean, then 47.72% of the data are between the mean
and that number. When we use the table, we are assuming that the data are normally distributed. Use the table above to determine the percentage of the data between x and the mean.
Area 10 Mathematics and Technology Professional Development Center Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999 |
