Interpreting the Standard Deviation

1
Interpreting the Standard Deviation

• Given two samples from a population, the sample
  with the larger standard deviation (SD) is the
  more variable
• Example from last time
• We are using the SD as a relative or comparative
  measure
• How does the SD provide a measure of variability
  for a single sample or, what does 29.6 really
  mean?

2
Interpreting the Standard Deviation (continued)

• Recall the list of numbers
• 10, 20, 30, 45, 50, 70, 85, 90
• How many measurements are within 1 SD, 2 SDs of
  the mean?

For 1 SD 4 out of 8, or 50
For 2 SD 8 out of 8, or 100
3
Chebyshevs Rule

• Applies to any data set, regardless of the shape
  of its frequency distribution
• No useful information on fraction of measurements
  falling within for samples and
  for populations
• At least of the measurements will fall w/in 2
  SD of the mean at least of the measurements
  will fall w/in 3 SD of the mean

4
Chebyshevs Rule (continued)

• General formulation
• For any number , at least of
  the measurements will fall within k SDs of the
  mean
• Gives the smallest percentages that are
  mathematically possible the observed percentages
  can be much higher

5
The Empirical Rule

• A rule of thumb that applies to data sets that
  have a mound shaped, symmetric distribution
• Approximately 68 of the measurements will fall
  within 1 SD of the mean
• Approximately 95 of the measurements will fall
  within 2 SDs of the mean
• Approximately 99.7 of the measurements will fall
  within 3 SDs of the mean

6
Numerical Measures of Relative Standing

• Descriptive measures of the relationship of a
  measurement to the rest of the data
• Percentile ranking---For any set of n ordered
  measurements, the pth percentile is a number such
  that p of the measurements fall below the pth
  percentile and (100-p) fall above it
• Example---Standardized tests in schools.
  Reported results often include percentile ranks.
  So your reading score was 119 and this
  corresponds to the 89th percentile 89 of the
  scores were below 119, 11 were above 119

7
Numerical Measures of Relative Standing
(continued)

• The z-score---specifies the relative location of
  an observation in a data set relative to the mean
  and SD of the data set represents the distance
  between a given measurement y and the mean,
  expressed in SDs
• Sample z-score
• Population z-score

8
Numerical Measures of Relative Standing
(continued)

• A large z-score indicates that the measurement is
  larger than almost all other measurements in the
  population or sample
• A large negative z-score indicates that the
  measurement is smaller than almost all other
  measurements in the population or sample

9
Interpretation of z-Scores for Mound-shaped
Distributions of Data

• Approx. 68 of the measurements will have a
  z-score between 1 and 1
• Approx. 95 of the measurements will have a
  z-score between 2 and 2
• Approx. 99.7 of the measurements will have a
  z-score between 3 and 3

10
Interpretation of z-Scores for Mound-Shaped
Distributions of Data (continued)

This interpretation is identical to that given by
the empirical rule The statement that a
measurement falls within the interval
or the interval is
equivalent to the statement that a measurement
has a population (or sample) z-score between 2
and 2
11
Distribution for measurements from a normal
population with

z-scale
-3 -2 -1 0 1 2
3
2.5 16 50 66
97.5 percentile scale
12
Interpretation of z-Scores for Mound-Shaped
Distributions of Data (continued)