50 CHAPTER 3 Graphics and Summary Statistics
Chebyshev ’ s inequality : The interval [ μ − k σ , μ + k σ ] contains at
least 100(1 − 1 / k 2 )% of the distribution or data, where μ is the mean
and σ is the standard deviation. Compare this with the empirical rule.
Chebyshev ’ s inequality guarantees at least 0% within 1 standard devia-
tion of the mean (essentially guarantees nothing), while the empirical
rule gives 68%. Chebyshev ’ s inequality guarantees at least 75% with
2 standard deviations of the mean, while the empirical rule gives 95%.
Chebyshev ’ s inequality always guarantees lower percentages than the
empirical rule. This is because Chebyshev ’ s rule must apply to all dis-
tributions that have variances while the empirical rule applies only to
distributions that are approximately normally distributed.
3.9 EXERCISES
- What does a stem - and - leaf diagram show?
- What does a relative frequency histogram show?
- What is the difference between a histogram and a relative frequency
histogram? - How is a relative frequency histogram different from a cumulative relative
frequency histogram? - What portion of the data is contained in the box portion or body of a box -
and - whiskers plot? - When are pie charts better than bar charts?
- What relationship can you make to the three measures of location (mean,
median, and mode) for right - skewed distributions? - What is the relationship between these measures for left - skewed
distributions? - What is the defi nition of mean absolute error (deviation)?
- What is the defi nition of mean square error?
- Under what conditions does a probability distribution contain approxi-
mately 95% of its mass within 2 standard deviations of the mean?