Chapter 17: expected value and standard error for the sum ...

Chapter 17: expected value and standard error for the sum of the draws
from a box

Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
When we do this 10,000 times... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Expected value and standard error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Expected value 5
Expected value for sum of the draws, method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Expected value for sum of the draws, method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Formula for expected value of sum of the draws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Standard error 9

Standard error for the sum of the draws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Computing the SE for the sum of the draws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Example (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Example (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Short-cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Normal approximation 16

Use normal approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Example (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Example (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Classifying and counting 21

Replace tickets by 0s and 1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1

Context

s We’ll look at sum of the draws of a box
s Example:

x Count the number of heads in 100 coin tosses
x Maybe one time the number is 54, the next time it is 48, the third time it is 47. The observed

value varies!
x Observed value = expected value + chance error
x See computer simulation, where I repeated this 10,000 times

2 / 22

When we do this 10,000 times...

Number of heads in 100 coin tosses, repeated 10000 times

0.08

0.06

Density
0.04

0.02

0.00

30 40 50 60
nr of heads

3 / 22

Expected value and standard error

s Note that the number of heads is a random variable, with a distribution!
s What is the center and spread of this distribution?

x The center is called the expected value
x The spread is called the standard error. The standard error gives the likely size of the chance error.
s We can use a similar model to analyze election polls, and will look into that later.

4 / 22

2

Expected value 5 / 22

Expected value for sum of the draws, method 1 6 / 22

s We look at the sum of 100 draws from a box with the tickets 0, 1, 1, 6
s Observed value = expected value + chance error
s What is the expected value of the sum of the draws?
s Method 1:

x How many 0’s do we expect in our draws? About 25.
x How many 1’s do we expect in our draws? About 50.
x How many 6’s do we expect in our draws? About 25.
x So what do we expect for the sum of the draws? About

(25 × 0) + (50 × 1) + (25 × 6) = 0 + 50 + 150 = 200

Expected value for sum of the draws, method 2

s Method 2:

x The average of the box is: 0 + 1 + 1 + 6 8
4 4
= = 2

x So after each draw, we expect the sum of the draws to increase by about 2

x So the sum of the draws is expected to be 100 × 2 = 200

x General formula for the expected value for the sum of the draws, made at random with
replacement:
(number of draws) × (average of the box)

7 / 22

Formula for expected value of sum of the draws

s General formula for the expected value for the sum of the draws, made at random with replacement:

(number of draws) × (average of the box)

s Does the formula make sense?
x What happens if the number of draws is doubled? Then the expected value of the sum of the
draws doubles.
x What happens if the average of the box is doubled? Then the expected value of the sum of the
draws doubles.
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3

Standard error 9 / 22

Standard error for the sum of the draws

s We look at the sum of draws from a box
s Observed value = expected value + chance error
s How big is the chance error? The chance error is likely to be similar in size to the standard error (SE)

for the sum of the draws
s If the SE for the sum of the draws is large, then we have large chance errors, and the observed values

are widely spread around the expected value
s If the SE for the sum of the draws is small, then we have small chance errors, and the observed values

are tightly clustered around the expected value
s Observed values are rarely more than 2 or 3 SEs away from the expected value.

10 / 22

Computing the SE for the sum of the draws

√
s SE for the sum of the draws = number of draws × (SD of the box)
s This is called the square root law, because it involves the square root of the number of draws
s Does the formula make sense?

x What happens if the n√umber of draws is doubled? Then the SE of the sum of the draws is
multiplied by a factor 2. This matches with what we learned about the law of large numbers:
the chance error grows, but only slowly.

x What happens if we double the SD of the box? Then the SE of the sum of the draws doubles.

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Example

s We look at the sum of 25 draws from a box with tickets 0,2,3,4,6

s Fill in the blank. The sum of the draws is around ...(a), give or take ...(b) or so.

s (a) should be the expected value of the sum of the draws:

(number of draws) × (average of the box)

= 25 × 0+2+3+4+6 = 25 × 3 = 75
5

s (b) should be the SE for the sum of t√he draws.
This is given by the square root law: number of draws × (SD of the box)

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4

Example (cont’d)

s W√e need to compute the SE for the sum of the draws:
number of draws × (SD of the box)

s What is the SD of the box 0, 2, 3, 4, 6?
x Step 1: compute the average of the box: 3 (see part a)
x Step 2: compute deviation from the average:
-3, -1, 0, 1, 3
x Step 3: compute r.m.s. size of the deviations:

(−3)2 + (−1)2 + 02 + 12 + 32 = 20 = √ = 2
5 5 4

x So the SD of the box is 2
s √The SE for the sum of the draws is:

25 × 2 = 5 × 2 = 10.

13 / 22

Example (cont’d)

s We look at the sum of 25 draws from a box with tickets 0,2,3,4,6
s Fill in the blank. The sum of the draws is around ...(a), give or take ...(b) or so.
s (a) should be the expected value of the sum of the draws: 75
s (b) should be the SE for the sum of the draws: 10
s So the sum of the draws is around 75, give or take 10 or so.

14 / 22

Short-cut

s Suppose the box only contains two kinds of tickets: some tickets with a big number and some tickets
with a small number. Then there is a shortcut to compute the SD of the box!

s SD of the box =
(big number − small number)
× (fraction of big numbers) × (fraction of small numbers)

s Example: box with tickets 7,7,7,-2,-2
x Large number = 7. Fraction of large numbers = 3/5.
x Small number = -2. Fraction of small numbers = 2/5.
x SD of the box = (7 − (−2)) × (3/5) × (2/5) = 9 × (3/5) × (2/5)
x Use calculator to compute this

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5

Normal approximation 16 / 22

Use normal approximation

s If number of draws is large, we can use the normal approximation to estimate chances.
s We should use a new average and new SD:

x New average = expected value for sum of the draws
x New SD = SE for the sum of the draws
x So the new standard units tell us how many SEs a number is away from the expected value

17 / 22

Example

s Consider the sum of 25 draws from the box with tickets 0,2,3,4,6.
s See computer simulation, where I repeated this 1000 times

18 / 22

Example (cont’d)

Histogram of sum of the draws, when repeated 1000 times

0.04

0.03

Density
0.02

0.01

0.00

40 50 60 70 80 90 100 110 19 / 22
sum of the draws

6

Example (cont’d)

s About what percentage of observed values should be between 50 and 100?
s We use the normal approximation:

x New average: expected value for the sum of the draws = 75
x New SD: SE for the sum of the draws = 10
s Note that these numbers match with the graph on the previous slide.
s Then use normal approximation as before. See overhead

20 / 22

Classifying and counting 21 / 22

Replace tickets by 0s and 1s

s See overhead for example
s Suppose you draw from a box, and want to count the number of a certain ticket (or tickets)
s Then:

x put a 0 on the tickets that you don’t want to count
x put a 1 on the ticket that you do want to count
s Using the new box:
x The count is like the sum of the draws from the new box
x We can compute the expected value and SE as before
x We can also use the normal curve to approximate probabilities as before

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7