Transcript standard deviation - A Site for Mathematical Minds

Statistics
Describing, Exploring and Comparing Data
Chapter 3
Example 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.
2
• The formula for the sample variance is , 2  x  x
s 
n 1


– where x is each sample value, that is 55, 51, 64, 70 and 49 and
– x 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
Measures of Variation
• So the sample variance is taking each x value,
subtracting the mean, square this and
summing these up, that is
 55  57.8   51  57.8   64  57.8   70  57.8   49  57.8
s2 
2
2
2
2
2
5 1
– 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
Measures of Variation
s
2
 2.8

2
  6.8   6.2   12.2    8.8 
2
2
2
2
5 1

7.84  46.24  38.44  148.84  77.44 318.8

 79.7
4
4
– Remember to then get the sample standard
deviation to simply take the square root of 79.7
•
s  79.7  8.9 seconds
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
s  79.7  8.9