Download 2.4 Notes

2.4 Standard Deviation
AP Statistics
Review: What are the measures of central tendency?
New: The measure of spread are
,
Which has the larger spread? A or B?
A.
B.
Here are some symbols we use for the following:
in a sample:
in a population:
size:
mean:
variance:
standard deviation:
Definition: The
of an entry x in a population data set is the
difference between the entry and the mean µ of the data set.
** Deviation of x =
Guidelines for finding POPULATION Variance and Standard Deviation
∑𝑥
1. Find the mean of the population data set
𝜇=
𝑁
2. Find the deviation of each entry
x–µ
3. Square each deviation
(x – µ)2
4. Add to get the sum of squares
𝑆𝑆𝑥 = ∑(𝑥 − 𝜇)2
5. Divide by N to get the population
variance
𝜎2 =
6. Find the square root of the variance to get
the population standard deviation
∑(𝑥 − 𝜇)2
𝜎=√
𝑁
∑(𝑥 − 𝜇)2
𝑁
Example 1: Find the deviation of each starting salary (in 1000s of dollars) for Corporation A.
Salary
x
41
38
39
45
47
41
44
41
37
42
∑𝑥 =
Deviation
x–µ
Squares
(x – µ)2
This is the reason why we don’t
take the average of the
deviation … it would always be
∑(𝑥 − 𝜇) =
SSx =
. So we solve
that problem by squaring the
deviation, then taking the
average. Tricky, huh?
Example 2: Find the variance and standard deviation of the starting salaries for Corporation A.
* Our population of interest is Corporation A Salaries *
Guidelines for finding SAMPLE Variance and Standard Deviation
1. Find the mean of the population data set
2. Find the deviation of each entry
3. Square each deviation
4. Add to get the sum of squares
∑𝑥
𝑛
x – 𝑥̅
(x – 𝑥̅ )2
𝑥̅ =
𝑆𝑆𝑥 = ∑(𝑥 − 𝑥̅ )2
∑(𝑥 − 𝑥̅ )2
𝑛−1
5. Divide by n – 1 to get the sample variance
𝑠2 =
6. Find the square root of the variance to get the sample
standard deviation
∑(𝑥 − 𝑥̅ )2
𝑠=√
𝑛−1
Example 3: Find the sample variance and standard deviation of the starting salaries for
Corporation A.
* Our population of interest is ALL corporations. *
AP Statistics
2.4 Measures of Variation (continued)
sample
population
sample size
mean
standard deviation
One of the most important distribution shapes is the
Bell-Shaped distributions are important because we can use the
to determine some characteristics about the data.
Empirical Rule: In data with a bell shaped distribution, the data will have the following
characteristics:
About
of the data lies within 1 standard deviation of the mean.
About
of the data lies within 2 standard deviation of the mean.
About
of the data lies within 3 standard deviation of the mean.
Example: In a sample of 10,000 SAT test scores the data was approximately bell shaped with
mean 1400 and standard deviation 140.
1.) About how many students scored between 1260 and 1540?
2.) About how many students scored between 980 and 1680?
3.) How many students scored below 1120?
4.) What must you score to be in the top 16% of students?
5.) Tonya was ill on test day. She scored in the bottom 2.5% of scores.
Her score was at most
.
AP Statistics
2.4 Measures of Variation (continued)
The EMPIRICAL RULE applies to (symmetric) bell-shaped distributions.
What if it’s not bell-shaped?????
Use
!
Chebychev’s Theorem: The portion of any data set lying within k standard deviations (k > 1) of
the mean is at least
***This gives a MINUMUM percent of data values that fall within the given number of standard
deviations of the mean.***
Example: The age distributions for Alaska and Florida are shown in the histograms. Decide
which is which. Apply Chebychev’s Theorem to each data set using k = 2 and state your
conclusions.
STANDARD DEVIATION FOR GROUPED DATA:
The formula for sample standard deviation for a frequency distribution is
Sample standard deviation = s =
3) You collect a random sample of the number of children per household in a region. The results are
shown below. Find the sample mean and the sample standard deviation of the data set.
Number of Children
in 50 Households
1
1
1
1
3
1
3
2
4
0
3
2
1
5
0
1
6
3
1
3
1
2
0
0
3
6
6
0
1
0
1
1
0
3
1
0
1
1
2
2
1
0
0
6
1
1
2
1
2
4
Using Midpoints of Classes
4) The circle graph below shows the results of a survey in which 1000 adults were asked how much they
spend in preparation for personal travel each year. Make a frequency distribution for the data. Then
use the table to estimate the sample mean and the sample standard deviation of the data set.