statistics describing, exploring and comparing data chapter 3 example problems
TRANSCRIPT
StatisticsDescribing, Exploring and Comparing Data
Chapter 3Example Problems
Measures of Variation
• Objective: Find the range, variance and standard deviation of sample data
• Question: Statistics students participated in an experiment to test their ability to determine when I minute (or 60 seconds) has passed. The results are given below in seconds. – Find the range, variance, and standard deviation for the given
sample data. – Identify one reason why the standard deviation from this sample
might not be a good estimate of the standard deviation for the population of adults.
55 51 64 70 49
Measures of Variation
• Range: The difference between the high and low values– Range = High – Low = 70 – 49 = 21 seconds
55 51 64 70 49
Measures of Variation
• Sample variance: measure of variation of values (from a sample) from the mean– You want to learn with a small sample how to compute the sample
standard deviation for your understanding but then you want to move to technology as it will be faster. So let’s do this one both ways.
• The formula for the sample variance is ,
– where x is each sample value, that is 55, 51, 64, 70 and 49 and – is the mean of the sample, that is (55+51+64+70+49)/5 = 57.8 and – n is the number of samples which is 5.
55 51 64 70 49
22
1
x xs
n
x
Measures of Variation
• So the sample variance is taking each x value, subtracting the mean, square this and summing these up, that is
– As you can see already this is a lot of calculations but this is good to know that this is how the formula works. You can go through and simplify the parentheses, then square to get
55 51 64 70 49
2 2 2 2 22 55 57.8 51 57.8 64 57.8 70 57.8 49 57.8
5 1s
Measures of Variation
– Remember to then get the sample standard deviation to simply take the square root of 79.7
• seconds
2 2 2 2 22 2.8 6.8 6.2 12.2 8.8 7.84 46.24 38.44 148.84 77.44 318.8
79.75 1 4 4
s
79.7 8.9s
MEASURES OF VARIATION
• Technology Way• In your TI-83/84 you first have to put the values
in a list. – STAT / EDIT / 1:Edit Enter– Enter each of the values 55, 51, 64, 70, 49 (pressing
enter to move down after each one)– Then press STAT / CALC / 1-Var Stats / Enter and put
the L1 list you created by 2nd 1 and press Enter. • You will see the mean of 57.8 that we found and the 8.9
standard deviation.
55 51 64 70 49
55 51 64 70 49
79.7 8.9s