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Prob/Stat – Unit 5
Name: ________________________________________
Z-Scores Pre-Notes Fill-Out
Date: __________________________
Z-Scores (Standardizing Data)
Think About This: Both the SAT and ACT are used by colleges for admission, but is a 34 ACT better
than a 2300 SAT? Can you tell the differences between the scores?
To help figure this out we use something called:
Z-Scores (formula below)
𝑧=
π‘‘π‘Žπ‘‘π‘Ž π‘£π‘Žπ‘™π‘’π‘’ βˆ’ π‘šπ‘’π‘Žπ‘›
π‘₯βˆ’πœ‡
=
π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘›
𝜎
- Make sure to round your answers to 2 decimal values.
- A number representing the number of standard deviations from the mean (this can be positive or
negative – basically the distance)
- The mean always has a z-score of 0 (duh, it is a distance of 0 from the mean...)
ON YOUR OWN: Using your phone, Google search for when z-scores are unusual and very
unusual. Specify on the number line between what z-score values it is unusual and very unusual:
-3
-2
-1
0
1
2
3
For example:
On an English test, you scored an 85 when the average was 75 with a standard deviation of 8.
On a Math test, you scored an 88 when the average was 80 with a standard deviation of 6.
β€―a) Which test did you perform better on relative to your peers?
Here is how to answer this question using the z-score formula:
π‘¬π’π’ˆπ’π’Šπ’”π’‰ 𝑻𝒆𝒔𝒕: 𝑧 =
𝑴𝒂𝒕𝒉 𝑻𝒆𝒔𝒕: 𝑧 =
π‘‘π‘Žπ‘‘π‘Ž π‘£π‘Žπ‘™π‘’π‘’ βˆ’ π‘šπ‘’π‘Žπ‘›
π‘₯βˆ’πœ‡
85 βˆ’ 75
10
=
=
=
= 1.25
π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘›
𝜎
8
8
π‘‘π‘Žπ‘‘π‘Ž π‘£π‘Žπ‘™π‘’π‘’ βˆ’ π‘šπ‘’π‘Žπ‘›
π‘₯βˆ’πœ‡
88 βˆ’ 80
8
=
=
= = 1.33
π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘›
𝜎
6
6
Now Tell Me: Looking at both z-score values, which test did you perform better on?
(Think about it…the z-score tells you how far you are from the average.)
β€―b) Would you consider either score unusual? (Hint: Based on your Google search about this, which
of these z-scores fall in this category?)
YOUR TURN TO TRY!
Example 1: In a local city the electrical bills have a mean of $47 with a standard deviation of $8.
Find all three z-scores that correspond to $35, $47, and $66.
Example 2: Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math
scores had a mean 516 and standard deviation 114. Gerald takes the ACT Assessment mathematics
test and scores 27. ACT math scores in 2002 had a mean of 20.6 and standard deviation
5.0. Assuming that both tests measure the same kind of ability, who has the higher score? (Hint:
same as the Math and English Test example)
Example 3: The heights of women aged 20 to 29 have mean 64 inches and standard deviation 2.7
inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches. What
are the z-scores for a woman 6 feet tall and a man 6 feet tall?
Example 4: The Yotoya Pria is a hybrid car known to average 45 miles per gallon (mpg). This does
not mean all the cars will achieve 45 mpg though. My Pria has a z-score of -0.55 and I know the
standard deviation is 3.7 mpg. What is my car's mile per gallon? (Hint: Use the z-score formula to
plug in what you know. You might have to do some algebra to solve for a certain variable!!
Remember: When solving, always isolate the variable!!)
Example 5: One particular Super Bouncy Ball when dropped from the top of W-L bounced 24 feet in
the air. This bounce had a z-score of 2.37 and the standard deviation of all drops is known to be 5.8
feet. What is the mean bounce height of the Super Bouncy Balls? (Hint: Same as Example 4.)