Math 3
1-5 Standardized Values and Percentiles
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Name__________________________________
I can find and interpret standardized values
I can use the standard normal distribution to find probabilities and percentiles
1. Examine the chart to the right, which gives
approximate information about the heights of young
Americans ages 18 to 24 years old. Each distribution
is approximately normal.
1a.
Sketch the two distributions below.
1b.
What percentage of males age 18-24 are between 65.8 and 71.2 inches tall? _________
1c.
Jackie is 68 inches tall. What percentage of females are smaller than Jackie? ________
[The answer to this question is also called Jackie’s percentile height. Remember, a percentile is the
percentage of data at or below a given value or data point.]
2a.
If an 18-24 year old male is in the 16th percentile, how tall is he?
2b.
Vinnie’s height is 2.1 standard deviations below the mean. How tall is he?
2c.
Annabel is 74 inches tall. How many standard deviations above average height is she?
2d.
Courtney is 68 inches tall. Steve is 71 inches tall. Who is relatively taller for his or her gender? Explain
your reasoning.
The standardized value or z-score is the number of standard deviations a given value lies from the mean. For
example, in problem (1c), Jackie is 1 standard deviation above the mean, so her z-score is 1. In problem (2a),
Vinnie is 2.1 standard deviations below the mean, so his z-score is 2.1.
3a.
Compute the standardized value for Annabel from problem (2c).
3b.
What is a formula for calculating a z-score? Write it in terms of the data value x , the mean of the
data  , and the standard deviation of the data  . Check your answer with Mr. Sheppard before
moving on.
4a.
Compute the standardized value for a young woman who is 5' tall.
4b.
Compute the z-score for a young man who is 5'2 '' tall.
4c.
Who is taller for their gender, the young man or the young woman?
When a standardized value is 1 , 2 , or 3 standard deviations from the mean, it is very easy to use a normal
distribution to compute the probability that a value is more likely or less likely to occur than the standardized
value.
For instance, if a z-score is 1, then we can use a normal distribution to find P  z  1  0.84
[read: “the probability that z is less than 1 is 0.84”].
If a standardized value is 2, then we can use a normal distribution to find P  z  2  0.025
What if a standardized value is not a value between 3 and 3 ? How do we find probabilities then?
Take notes below:
5a.
Marcus is 74 inches tall. What is Marcus’ z-score? What is Marcus’ percentile height?
5b.
Joe is 62 inches tall. What is Joe’s percentile?
5c.
Steve is 70 inches tall. What percent of 18-24 year olds are taller than Steve?
5d.
What percent of 18-24 year old males are between Joe’s height and Steve’s height?
5e.
Silvia is at the 20th percentile in height. What is her height?
6.
The Wechsler Intelligence Scale for Children are (within each age group) approximately normally
distributed with a mean of 100 and a standard deviation of 15.
6a.
Draw a sketch of the distribution of these scores below. Be sure to scale and label the x-axis.
6b.
What percentage of children of a given age group have IQs above 150?
6c.
What IQ score would be at the 50th percentile?
6d.
If a student’s IQ is at the 75th percentile, what is the student’s IQ score?
7.
In 1996, the scaled scores on the SAT verbal section had a mean of 505 and a standard deviation of
110. Assume that the scaled scores are normally distributed. What is the probability that a
randomly selected student in 1996 had an SAT verbal score between 450 and 600?