Interpreting

1
Interpreting Center Variability
2
Strategy for Exploring Data

• Always plot your data make a graph, usually a
  histogram or a stemplot.
• Look for the overall pattern (shape, center,
  spread) and for striking deviations such as
  outliers.
• Calculate a numerical summary to briefly
  describe the center and spread.
• Sometimes the overall pattern of a large number
  of observations is so regular that we can
  describe it by a smooth curve.

3
Density Curves

• Can be created by
  smoothing histograms
• ALWAYS on or above
  the horizontal axis
• Has an area of exactly
  one underneath it
• Describes the proportion of
  
  observations that fall within a range
  of values
• Is often a description of the overall
  distribution
• Uses m s to represent the mean standard
  deviation
• Outliers are not described by a density curve
• An approximation that is easy to use and accurate
  enough for practical use.

4
The area of the shaded bars represents students
with scores 6.0 or lower which is equal to .303
of all students.
The shaded area under the curve represents the
proportion of students with scores 6.0 or lower
which is equal to .293, only .010 away from the
histogram itself.
5
Chebyshevs Rule
At least what percent of observations is within 2
standard deviations of the mean for any shape
distribution?

• The percentage of observations that are within k
  standard deviations of the mean is at least
• where k gt 1
• can be used with any distribution

75
6
Chebyshevs Rule- what to know

• Can be used with any shape distribution
• Gives an At least . . . estimate
• For 2 standard deviations at least 75

7
Normal Curve

• Bell-shaped, symmetrical curve
• Transition points between cupping upward
  downward occur at m s and m s
• As the standard deviation increases, the curve
  flattens spreads
• As the standard deviation decreases, the curve
  gets taller thinner

8
The exact density curve for a particular normal
distribution is described by giving its mean ?
and its standard deviation ?.
Standard deviation ? controls the spread of a
normal curve. The curve with the larger ? is
more spread out.
9

• Why are normal distributions important?
• Normal distributions are good descriptions for
  some distributions of real data.
• Normal distributions are good approximations to
  the results of many kinds of chance outcomes.
• Many statistical inference procedures based on
  normal distributions work well for other roughly
  symmetric distributions.

10
Empirical Rule

• Approximately 68 of the observations are within
  1s of m
• Approximately 95 of the observations are within
  2s of m
• Approximately 99.7 of the observations are
  within 3s of m

Can ONLY be used with normal curves!
11
The height of male students at SWVGS is
approximately normally distributed with a mean of
71 inches and standard deviation of 2.5 inches.
a) What percent of the male students are shorter
than 66 inches? b) Taller than 73.5 inches? c)
Between 66 73.5 inches?
About 2.5
About 16
About 81.5
12
Remember the bicycle problem? Assume that the
phases are independent and are normal
distributions. What percent of the total setup
times will be more than 44.96 minutes?
First, find the mean standard deviation for the
total setup time.
2.5
13
z score

• Standardized score
• Creates the standard normal density curve
• Has m 0 s 1

14
What do these z scores mean?

• -2.3
• 1.8
• 6.1
• -4.3

2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean
15
Jonathan wants to work at Utopia Landfill. He
must take a test to see if he is qualified for
the job. The test has a normal distribution with
m 45 and s 3.6. In order to qualify for the
job, a person can not score lower than 2.5
standard deviations below the mean. Jonathan
scores 35 on this test. Does he get the job?
No, he scored 2.78 SD below the mean