3. Describing Data in terms of the
Standard Deviation.
Test Mean = 80
St. Dev. = 5
4. Chebyshev’s Rule
The percent of observations that are within
k standard deviations of the mean is at
least
2
1
100 1 %
k
5. Facts about Chebyshev
Applicable to any data set – whether it is
symmetric or skewed.
Many times there are more than 75% - this
is a very conservative estimation.
6. # St. Dev. % w/in k st. dev. of mean
2
3
4
4.472
5
10
2
1
100 1
k
7. Interpret using Chebyshev
Test Mean = 80
St. Dev. = 5
1. What percent are between 75 and 85?
2. What percent are between 60 and 100?
8. Collect wrist measurements (in)
Create distribution
Find st. dev & mean.
What percent is within 1 deviation of mean
9. Practice Problems
1. Using Chebyshev, solve the following problem for a
distribution with a mean of 80 and a st. dev. Of 10.
a. At least what percentage of values will fall between 60
and 100?
b. At least what percentage of values will fall between 65
and 95?
10. Normal Distributions
These are special density curves.
They have the same overall shape
Symmetric
Single-Peaked
Bell-Shaped
They are completely described by giving its
mean () and its standard deviation ().
We abbreviate it N(,)
11. Normal Curves….
•Changing the mean without changing the standard
deviation simply moves the curve horizontally.
•The Standard deviation controls the spread of a Normal
Curve.
12. Standard Deviation
It’s the natural measure of spread for Normal
distributions.
It can be located by eye on a Normal curve.
It’s the point at which the curve changes from concave
down to concave up.
13. Why is the Normal Curve Important?
They are good descriptions for some real data
such as
Test scores like SAT, IQ
Repeated careful measurements of the same quantity
Characteristics of biological populations (height)
They are good approximations to the results of
many kinds of chance outcomes
They are used in many statistical inference
procedures.
14. Empirical Rule
Can only be used if the data can be
reasonably described by a normal curve.
Approximately
68% of the data is within 1 st. dev. of mean
95% of the data is within 2 st. dev. of mean
99.7% of data is within 3 st. dev. of mean
16. Empirical Rule (68-95-99.7 Rule)
In the Normal distribution with mean ()
and standard deviation ():
Within 1 of ≈ 68% of the observations
Within 2 of ≈ 95% of the observations
Within 3 of ≈ 99.7% of the observations
17. The distribution of batting average (proportion of hits) for the 432
Major League Baseball players with at least 100 plate appearances
in the 2009 season is normally distributed defined N(0.261, 0.034).
Sketch a Normal density curve for this distribution of batting
averages. Label the points that are 1, 2, and 3 standard
deviations from the mean.
What percent of the batting averages are above 0.329?
What percent are between 0.227 and .295?
18. Scores on the Wechsler adult Intelligence Scale (a
standard IQ test) for the 20 to 34 age group are
approximately Normally distributed. N(110, 25).
What percent are between 85 and 135?
What percent are below 185?
What percent are below 60?
19. 2. A sample of the hourly wages of employees who work in
restaurants in a large city has a mean of $5.02 and a st.
dev. of $0.09.
a. Using Chebyshev’s, find the range in which at least
75% of the data will fall.
b. Using the Empirical rule, find the range in which at
least 68% of the data will fall.
20. The mean of a distribution is 50 and the standard deviation is
6. Using the empirical rule, find the percentage that will fall
between 38 and 62.
21. A sample of the labor costs per hour to assemble a certain
product has a mean of $2.60 and a standard deviation of
$0.15, using Chebyshev’s, find the values in which at least
88.89% of the data will lie.
