4.4 Interpreting Center and
Variability
Thursday, July 13, 2017
Mean and Std. Deviation can be
combined to obtain informative
statements about:
1. How the values in the data set are
distributed
2. About the relative position of a particular
value in a data set.
We do this by describing how far away a
particular observation is from the mean in
terms of the standard deviation.
For example, we might say “I scored two
standard deviations above the mean on
the last test!”
Chebyshev’s Rule (not on test)
1I
F
100G
1 J
%
Hk K
2
Percentage of observations
(data) that are within k std. dev.
of the mean
Applicable to any data set, symmetric or
skewed
1 % w/i k std. dev.
k (# of std.
1 2
dev.)
k of mean
1
2
1  .75 At least 75%
4
3
1
1  .89 At least 89%
9
4
1
1  .94
16
At least 94%
The Empirical Rule
Can be applied whenever the distribution of
data can be reasonably described by a
normal curve or bell-shaped (not for just
any data set)
 68% of data within 1 std. dev of mean
 95% of data within 2 std. dev of mean
 99.7% of data within 3 std. dev of mean
Dividing percentages in half is permissible because the shape
is symmetric
Understand that it is pretty unusual to see an observation, from
a normal dist., that is farther than 2 std. devs. from the mean.
Measures of Relative Standing –
z scores
z scores measure how many standard
deviations away from the mean a
particular piece of data is
value - mean
z score =
std. dev.
This process can be called standardization
and a z score is an example of a
standardized score
Z scores can be particularly useful when the
distribution of data is approx. normal in shape.
A percentile gives the percentage of data that falls
below a particular observation
If you score a 650 and are told you are in the 95th
percentile, 95% of the rest of the scores who
tested with you are below you.
The median marks the 50th percentile
The lower quartile marks the 25th percentile
The upper quartile marks the 75th percentile
Example
The SAT reasoning Test has 3 parts: Writing, Math,
and Verbal. Each part has a distribution that is
approximately normal. It is designed to have an
overall mean of about 500 and a standard
deviation of 100 for all test takers.
Question: Suppose you earned a 600 on one part
of your SAT. Where do you stand among all
students who took the test?
What about a score of 700?