Chapter 16 Part 3
MORE ABOUT MEANS AND VARIANCES
Adding a constant to data (adding the same amount
to each value) shifts the mean but the variance and
standard deviation do not change.
𝐸 𝑋±𝑐 =𝐸 𝑋 ±𝑐
𝑉𝑎𝑟 𝑋 ± 𝑐 = 𝑉𝑎𝑟 𝑋
𝑆𝐷 𝑋 ± 𝑐 = 𝑆𝐷 𝑋
Example:
Couples at the Quiet Nook can expect discounts averaging $5.83
with a standard deviation of $8.62. Suppose that for several
weeks the restaurant has also been distributing coupons worth
$5 off any one meal per table. If every couple there brings a
coupon, what will be the mean and standard deviation of the total
discounts they’ll receive?
E(x) = 5.83 + 5 = $10.83
SD(x) = $8.62 (unchanged)
Multiplying each value of a random variable by
a constant multiplies the mean and the
standard deviation by that constant.
𝐸 𝑎𝑋 = 𝑎 ∙ 𝐸(𝑋)
𝑆𝐷 𝑎𝑋 = 𝑎 ∙ 𝑆𝐷(𝑋)
Example: (continued)
Couples at the Quiet Nook can expect discounts averaging
$5.83 with a standard deviation of $8.62. When two
couples dine together on a single check, the restaurant
doubles the discount offer - $40 for the ace of hearts on
the 1st draw or $20 for ace of hearts on 2nd draw. What are
the mean and standard deviation?
E(x) = 2(5.83) = $11.66
StandardDev(x) = 2(8.62) = $17.24
Example:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
2Y + 20
Mean
Deviation
Mean = E(2Y+20) =
2E(Y)+20 = 2(12)+20 = 44
X
80
12
SD = SD(2Y+20) = 2SD(Y)
= 2(3) = 6
Y
12
3
Adding/Subtracting Random Variables:
𝐸 𝑋 ± 𝑌 = 𝐸(𝑋) ± 𝐸(𝑌)
If X and Y are independent, then
𝑉𝑎𝑟 𝑋 ± 𝑌 = 𝑉𝑎𝑟 𝑋 + 𝑉𝑎𝑟 𝑌
𝑆𝐷 𝑋 ± 𝑌 =
𝑆𝐷(𝑋)2 + 𝑆𝐷(𝑌)2
Example:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
X+Y
Mean
Deviation
Mean = E(X+Y) =
E(X)+E(Y)= 80+12= 92
X
80
12
SD = SD(X+Y)
= 122 + 32 = 12.37
Y
12
3
Practice:
Given independent random variables with means and
standard deviations as shown, find the mean and
standard deviation of:
Standard
a) 3X
b) 0.25X+Y
c) X – 5Y
d) 𝑋1 +𝑋2 + 𝑋3
Mean
Deviation
X
80
12
Y
12
3
Answers:
a)240; 36
b)32; 4.24
c)20; 19.21
d)240; 20.78
Today’s Assignment:
Add to HW #9 p. 384 #24-26, 33-36