Standard Deviation
Essential Understanding:
~ Variance and Standard deviation are measures showing how much data values deviate from the mean.
~The Greek letter σ (sigma) represents standard deviation
~ σ2 ­ (sigma squared) is the variance ­­­in calculator you take σx and square it
Key Concepts:
~ x: Symbol for mean (the average of the data)
~ σx: Symbol for standard deviation in calculator
~ ox2: How you find variance in the calculator
Problem 1: Finding the mean, variance and standard deviation
What are the mean, variance, and standard deviation of these values?
6.9 8.7 7.6 4.8 9.0
Calculator (to find mean, standard deviation)
Step1: STAT EDIT enter data in list L1
Step2: STAT CALC slect the 1­Var Stats option
x: mean and σx: is standard deviation
Calculate Variance
Step3: VARS, #5 Statistics
Step4: #4 σx and square it Problem 2: The table displays the number of U.S. hurricane strikes by decade from the years 1851 to 2000. What are the mean, variance, and standard deviations?
decad
e
1
strike
s
19 15 20 22 21 18 21 13 19 24 17 14 12 15 14
(Mean) x:
2
3
4
5
6
7
(standard deviation) σx:
8
9
10 11 12 13 14 15
(variance) σx2:
1
Problem 3: The table displays the number of U.S. hurricane in the Atlantic Ocean from 1992 to 2006. What are the mean, variance, and standard deviations?
decade
1
2
3
4
strikes
4
4
3
11 10 3
(Mean) x:
5
6
8
9
10 11 12 13 14 15
10 8
8
9
7
(standard deviation) σx:
4
7
9
14 5
(variance) σx2:
Understand Normal Bell Curve with Standard Deviation
2
2
4
5
6
7
9
10 11 12 13 14 15
1
strikes
19 15 20 22 21 18 21 13 19 24 17 14 12 15 14
(Mean) x:
3
8
decade
(standard deviation) σx:
(variance) σx2:
Using Standard Deviation to Describe Data. Using data from Problem #2 (above)
How many standard deviations from the mean do all the values fall?
How many values fall within one standard deviations from the mean?
How many values fall within two standard deviations from the mean?
3
4
An entire distribution can often be reduced to a mean and standard deviation. A location of an individual score. z­score uses that information to indicate the Essentially, z­scores indicate how many standard deviations you are away from the mean. If z = 0, you’re at the mean. If z is positive, you’re above the mean; if negative, you’re below the mean. In practical terms, z scores can range from ­3 to +3.
z­score= x(score)­mean/standard deviation
Data was collected outside a popular new restaurant to determine the mean waiting time to be seated at a table. Assume the data was normally distributed with a mean of 45 minutes and a standard deviation of 12 minutes.
a) Determine the probability that a randomly selected person has to wait less than 20 minutes. Express to nearest tenth of a percent.
b) Determine the probability that a randomly selected person has to wait more than 1 hour. Express the the mearest tenth of a percent.
a) Find the z­score. z = (20­45)/12 = ­2.083
Next go to a standard normal table (probably at the end of your textbook and look up the probability. Or if you have a graphing calculator, such as a TI­83, you can type in normalcdf(­
1000,20,45,12) to get a probability of 0.0186.
b) z = (60­45)/12 = 1.25
probability = 0.1056
A normal distribution has a mean of 120 and a standard deviation of 20. For this distribution
a. What score separates the top 40% of the scores from the rest?
125
b. What score corresponds to the 90th percentile?
145.60
c. What range of scores would form the middle 60% of this distribution?
103.2 ù 136.80
5
6
7
z score example problem using z score table
1. A forester measured 27 of the trees in a large woods that is up for sale. He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches. Suppose that these trees provide an accurate description of the whole forest and that a normal model applies.
a. what size would you expect the central 95% of all trees to be?
b. about what percent of the trees should be less than an inch in diameter?
c. about what percent of the trees should be between 5.7 and 10.4 inches in diameter?
d. about what percent of the trees should be over 15 inches in diameter
a) what size would you expect the central 95% of all trees to be?
since the normal distribution is symmetric
we need to take 0.95/2 =0.475 on each side
so on the right hand side of the mean
1.96 = x­10.4/4.7
x= (4.7*1.96)+10.4 = 19.61
On the left hand side of the mean
­1.96 = x­10.4/4.7
x=10.4­(4.7*1.96) =1.18
so 95% is between 1.18 inches and 19.61 inches
b)
If the mean is 10.4 inch diameter, and the standard deviation is 4.7 inch diameter, then how many standard deviations is 1 inch diameter from the mean?
10.4 ­ z*4.7 = 1
z score = 2
So, in a normal distribution, what percentage of the results lie outside of 2 standard deviations from the mean (less than 1 inch diameter)? Well, there’s a formula that involves some pretty tough calculus, but luckily, it’s been tabulated.
Usually, tables will tell you the probability that a value lies WITHIN +­ z standard deviations of the mean. If you look up z = 2 in one of those tables, you’ll see that the probability is 95.45%. Since the probability is exactly 100% that the value lies either inside or outside of the range, that means the probability that it’s outside is
100 ­ 95.45 = 4.55%
Now, “outside” includes both above and below the range, and we just want the probability that the diameter is below 2 standard deviations from the mean. Since the normal distribution curve is symmetric, that means half the probability comes from above, and half from below. So, half of 4.55% is
4.55 / 2 = 2.28%
c. about what percent of the trees should be between 5.7 and 10.4 inches in diameter?
Z score= (5.7­10.4)/4.7= ­1
so percentage from z=­1 is 0.3413 or 34.13%
d)about what percent of the trees should be over 15 inches in diameter
z score= (15­10.4)/4.7 = 0.978
percent for trees over 15 inches in dia is from z table(0.978 ) is 0.3365 or 33.65%
8
http://mathbits.com/MathBits/TISection/Statistics2/normaldistribution.htm
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