STAT5002 Chap.7 The Box Model: Expected Value and Standard Error
The Box Model: Expected Value and Standard Error
Think of the box model as the simplest way to reason about randomness in an introductory statistics unit: imagine tickets in a box, each carrying a number, and a sample of tickets drawn at random. That picture turns two familiar summaries into new tools. The mean of the box becomes the expected value of a draw, written E(X), and the SD of the box, written σ, becomes its standard error, SE(X).
You will see how a single draw splits into its expected value plus a chance error, and why the standard error measures the typical size of that error rather than a hard limit. The chapter then gives you the computing formula for the SD of a box, mean of the squares minus the squared mean, which makes boxes with only two kinds of ticket, including the 0-1 box, a one-line calculation.
It also covers when a normal curve with the box mean and SD gives a sensible approximation for one draw, and why it can miss the exact count. Next comes the rule for adding draws: expected values add, and squared standard errors add, so plain SEs never do.
That rule, repeated for n draws with replacement, gives the four results you will use for the rest of the unit: the sum has expected value nμ and standard error σ√n, while the average has expected value μ and standard error σ/√n. Because later chapters on intervals and tests reuse exactly these formulas, getting comfortable here pays off for the rest of the semester.
Each idea comes with a fully computed example, a figure drawn from real numbers, and practice questions with complete answers, so you can check every step of your own working.
What this chapter covers
- 01
Box model: tickets, draws and random samples
- 02
Distribution of a single draw with unequal chances
- 03
Expected value E(X) as the mean of the box
- 04
Standard error SE(X) as the SD of the box
- 05
Chance error and the error box
- 06
Computing formula: mean square minus squared mean
- 07
Two-value boxes and the 0-1 box
- 08
Normal approximation for one draw with pnorm
- 09
Sums of two draws and the outer() function
- 10
Expected value and SE of a sum of n draws
- 11
Expected value and SE of a sample average
Worked example · free
Sum and average of sixteen draws
- +1Mean of the box: μ = (4 + 4 + 7 + 9)/4 = 6.
- +1Mean square = (16 + 16 + 49 + 81)/4 = 40.5, so σ = √(40.5 − 6²) = √4.5 = 2.121.
- +1Sum: E(S) = nμ = 16 × 6 = 96.
- +1Sum: SE(S) = σ√n = 2.121 × √16 = 8.485.
- +1Average: E(X̄) = μ = 6 and SE(X̄) = σ/√n = 2.121/4 = 0.530.
Key terms
- Box model
- A box model is a simple statistical model in which numbered tickets sit in a box and a sample is drawn at random, with or without replacement.
- Expected value
- The expected value E(X) is the mean of the box a random draw comes from; for a sum of n draws it is nμ.
- Standard error
- A standard error is the SD of the box of all possible values of a random quantity, giving the likely size of its chance error.
- Chance error
- A chance error is the random part of a draw, X − E(X), which behaves like one draw from an error box with mean zero.
- Mean square
- The mean square is the average of the squared tickets; subtracting the squared mean from it gives the squared SD of the box.
- 0-1 box
- A 0-1 box is a box containing only tickets marked 0 and 1, so its mean is the proportion p of ones and its SD is √(p(1 − p)).
- Sample average
- The sample average is the sum of n draws divided by n; with replacement it has expected value μ and standard error σ/√n, which shrinks as n grows.
The Box Model: Expected Value and Standard Error FAQ
What is the difference between standard deviation and standard error?
The SD describes the spread of the tickets in a box or of a list of data. The standard error describes the likely size of the chance error in a random quantity such as a draw, a sum or an average. For one draw they are numerically equal, but they answer different questions.
Why do you add squared standard errors instead of standard errors?
When two independent draws are added, high values from one box are often offset by low values from the other, so the spread of the sum is smaller than the two SEs added together. Working through all possible sums shows the squared SEs add exactly, so you square, add and take the root.
Does the standard error of an average get smaller with more draws?
Yes. The SE of an average is σ divided by the square root of n, so quadrupling the number of draws halves it. For the sum, the SE moves the other way and grows with the square root of n.
How do I find the SD of a box with only two kinds of ticket?
Use the computing formula: find the mean, find the mean of the squared tickets, subtract the squared mean and take the square root. For a 0-1 box this simplifies to the square root of p times one minus p.
Exam move
Start by building three small boxes of your own, one with equal chances, one with repeated tickets and one 0-1 box, and compute E(X) and SE(X) for each twice: once with the deviations and once with mean square minus squared mean. If the two answers differ, you have used the wrong divisor.
Next, take two tiny boxes, list every pair in R with outer(), and confirm that expected values add and squared SEs add; this one exercise makes the rule stick. Then write the four formulas for the sum and the average on one card and practise reading questions to decide whether they ask about a total or an average, since that choice decides between σ√n and σ/√n.
Finally, for each answer, write one sentence describing the result as a value give or take one SE, because interpreting the number is part of every box model question.
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