If XX is a normally distributed random variable and XX~N(μ, σ)N(μ, σ), then the z-score is:
z
=
x
-
μ
σ
z=
x
-
μ
σ
(1)
The z-score tells you how many standard deviations that the value xx is above (to the
right of) or below (to the left of) the mean, μμ. Values of xx that are larger than the mean
have positive z-scores and values of xx that are smaller than the mean have negative z-scores.
Suppose XX ~ N(5, 6)N(5, 6). This says that XX is a normally distributed random
variable with mean μ = 5μ = 5 and standard deviation σ = 6σ = 6. Suppose x = 17x = 17. Then:
z
=
x
-
μ
σ
=
17
-
5
6
=
2
z=
x
-
μ
σ
=
17
-
5
6
=2(2)
This means that x = 17x = 17 is 2 standard deviations (2σ)(2σ)
above or to the right of the mean
μ = 5μ = 5. The standard deviation is σ = 6σ = 6.
Notice that:
5
+
2
⋅
6
=
17
(The pattern is
μ
+
z
σ
=
x
.
)
5+2⋅6=17(The pattern isμ+zσ=x.)
(3)
Now suppose
x=1x=1. Then:
z
=
x
-
μ
σ
=
1
-
5
6
=
-
0.67
(rounded to two decimal places)
z=
x
-
μ
σ
=
1
-
5
6
=-0.67 (rounded to two decimal places)
(4)
This means that x = 1x = 1 is 0.67 standard deviations (- 0.67σ)(- 0.67σ) below or to the left of
the mean μ = 5μ = 5. Notice that:
5
+
(
-0.67
)
(
6
)
5+(-0.67)(6)
is approximately equal to 1
(This has the pattern
μ
+
(
-0.67
)
σ
=
1
μ+(-0.67)σ=1
)
Summarizing, when zz is positive, xx is above or to the right of μμ and when zz is negative, xx is to
the left of or below μμ.
Some doctors believe that a person can lose 5 pounds, on the average, in a
month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has a
normal distribution. Let XX = the amount of weight lost (in pounds) by a person in a month.
Use a standard deviation of 2 pounds. XX~N(5, 2)N(5, 2). Fill in the blanks.
Suppose a person lost 10 pounds in a month. The z-score when x = 10x = 10 pounds is z = 2.5z = 2.5
(verify). This z-score tells you that x = 10x = 10 is ________ standard deviations to the ________
(right or left) of the mean _____ (What is the mean?).
This z-score tells you that x = 10x = 10 is 2.5 standard deviations to the right
of the mean 5.
Suppose a person gained 3 pounds (a negative weight loss). Then zz = __________. This
z-score tells you that x = -3x = -3 is ________ standard deviations to the __________ (right or left)
of the mean.
zz = -4. This
z-score tells you that x = -3x = -3 is 4 standard deviations to the left
of the mean.
Suppose the random variables XX and YY have the following normal distributions:
XX ~ N(5, 6) N(5, 6) and Y ~ N(2, 1)Y ~ N(2, 1). If x = 17x = 17, then z = 2z = 2. (This was previously shown.)
If y = 4y = 4, what is zz?
z
=
y
-
μ
σ
=
4
-
2
1
=
2
where μ=2 and σ=1.
z=
y
-
μ
σ
=
4
-
2
1
=2where μ=2 and σ=1.(5)
The z-score for y = 4y = 4 is z = 2z = 2. This means that 4 is z = 2z = 2 standard deviations to the right of the
mean. Therefore, x = 17x = 17 and y = 4y = 4 are both 2 (of their) standard deviations to the right of
their respective means.
The z-score allows us to compare data that are scaled differently. To understand the
concept, suppose
XX ~ N(5, 6) N(5, 6) represents weight gains for one group of people who are trying
to gain weight in a 6 week period and YY ~ N(2, 1) N(2, 1) measures the same weight gain for a
second group of people. A negative weight gain would be a weight loss.
Since x = 17x = 17 and y = 4y = 4 are each 2 standard deviations to the right of their means, they
represent the same weight gain in relationship to their means.
"This is the course textbook for Biology 502 at CSU Dominguez Hills"