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Algebra 1 Notes
Z-Scores
The Normal Distribution
Z-scores
• z-scores allow for comparison of data that
are measured in different units.
• The comparison comes from standardizing
the data in terms of the standard deviation
and the mean.
z-scores
• Formula:
z
x-

Questions about z-scores
• Can a z-score be negative?
• Can a z-score be zero?
Interpeting Z-scores
The heights of 16 year-old males are normally
distributed with mean 68 inches and standard
deviation 2 inches.
Determine the z score for a male who is 66 inches
tall.
Facts: x = 66
 = 68
 =2
x
z

66 - 68

2
= -1
This height is 1 standard deviation below the mean.
Interpeting Z-scores
Jack scored 27 on the mathematics portion of the
ACT. If the math scores on the ACT are
normally distributed with mean of 20.7 and
standard deviation of 5.0, determine Jack’s zscore.
 = 20.7
Facts: x = 27
 = 5.0
x
z

27 - 20.7

5 .0
= 1.26
This score is 1.26 standard deviations above the mean.
Interpeting Z-scores
Jill scored 680 on the mathematics portion of the
SAT. If the math scores on the SAT are
normally distributed with mean of 518 and
standard deviation of 114, determine Jill’s z
score.
 = 518
Facts: x = 680
 = 114
x
z

680 - 518

114
= 1.42
This score is 1.42 standard deviations above the mean.
Interpreting z-scores
Assume the SAT and the ACT measure the
same kind of ability.
Who did better…Jack or Jill?
Since z = 1.42 is greater than z = 1.26, Jill’s
score of 680 on the SAT is relatively
“higher” than Jack’s score of 27 on the
ACT. Jill did better.
Interpreting z-scores
• The mean time it takes runners to
complete a cross-country race is 85
minutes. The standard deviation of
running times is 5 minutes.
• A runner has a z-score of -2. What does
this mean?
Interpreting z-scores
• The runner ran 2 minutes faster than the
overall mean.
• The runner ran 2 minutes faster than his
personal mean.
• The runner ran 2 standard deviations
faster than the overall mean.
• The runner ran 2 standard deviations
faster than his personal mean.
Consider the data shown below. This histogram shows the
heights of 1000 eighteen year old girls at a college campus
and the frequency with which those heights occurred.
What do you notice about the heights of the girls?
If we represented the data with a line plot, by connecting the
center of each bar of the histogram, our plot would look like
this:
Notice the general shape of this curve. Curves that have
this shape are often referred to as bell-shaped.
This curve can be smoothed even further:
This continuous smoothed curve is known as a normal curve.
This normal curve is part of a family of curves known as the
normal distribution.
The 68-95-99.7 Rule for Normal Distributions
Approximately 68% of the observations fall within 1
standard deviation of the mean
Approximately 95% of the observations fall within 2
standard deviations of the mean
Approximately 99.7% of the observations fall within 3
standard deviations of the mean
Another way of looking at it!