Where this week is heading

We build one connected story rather than five loose topics:

1. Variable types & frequency — what kind of data we have, and how we count it.
2. Population vs sample — the quantity we care about (\(\mu\)) versus the number we can actually compute (\(\bar{x}\)).
3. The mean — the estimator itself, its formula, and computing it in R.
4. Sampling for a fair estimate — how to sample so \(\bar{x}\) is a trustworthy estimate of \(\mu\).
5. Visualising discrete data in R — frequency tables, needle charts, and bar charts.

1.1 Types of variable

Every variable falls into one of two families, and each family splits in two:

1.2 Self-check — classifying variables

Click the statement you think is false. The answer and reasoning appear below.

1.3 Self-check — continuous data

1.4 Frequency, relative frequency & proportion

Common trap: writing 375/1081 (satisfactory ÷ unsatisfactory). The denominator is always the total.

1.5 Self-check — relative frequency

2.1 Population, sample, variable

A population can be:

• Finite and real — e.g. all KBS students enrolled in 2025.
• Effectively infinite or hypothetical — e.g. "all 2-litre bottles this filling process would ever produce on its current settings." We can never list them all.

2.2 Parameter vs statistic: \(\mu\) and \(\bar{x}\)

3.1 The mean as a measure of centre

Understanding the centre of a dataset is fundamental. The most common measure of centre is the mean — the average value.

3.2 Formula for the mean

Data are written \(\{x_1, x_2, \ldots\}\), or compactly \(\{x_i\}\). The choice of letter is arbitrary; \(x\) is conventional.

3.3 The mean in R

Manual averaging gets tedious as data grow. R has a built-in mean().

# A dataset of ten values
example_data <- c(24, 16, 30, 10, 12, 28, 38, 2, 4, 36)

# Calculate the mean
example_average <- mean(example_data)
print(paste("The average is:", example_average))   # 20

R also ships with datasets you can use directly:

data(iris)                                # load the built-in Iris data
mean_sepal_length <- mean(iris$Sepal.Length)
print(paste("Average sepal length is", mean_sepal_length))

3.4 A key property: deviations sum to zero

Subtract the mean from every value (this is mean-corrected or centred data). The corrected values always sum to exactly zero:

data <- c(4, 6, 2, 8)
mean_value <- mean(data)          # 5
data_mc <- data - mean_value      # -1  1  -3  3
sum(data_mc)                      # 0

3.5 Handling missing values

By default mean() returns NA if any value is NA. The argument na.rm = TRUE tells R to drop missing values first.

# Step 1: create the vector
data <- c(12, 18, NA, 14, 20, NA, 16, 10)

# Step 2: mean, ignoring missing values
mean_data <- mean(data, na.rm = TRUE)

# Step 3: print the result
print(paste("The mean value is:", mean_data))   # 15

The six present values are 12, 18, 14, 20, 16, 10 — sum 90, count 6, mean 15.

4.1 Simple random sampling (SRS)

Give every one of the \(N\) population members a unique ticket. Mix the tickets thoroughly in a hat and draw \(n\) of them — on a computer, we do this with random numbers.

4.2 Bias and convenience sampling

4.3 Self-check — the post-office queue

4.4 What SRS does — and does not — promise

• SRS guarantees fairness in the long run, not representativeness of one draw.
• Equal selection probability alone does not imply SRS was used — other schemes (e.g. stratified with proportional allocation, coming next) also give equal probabilities.

4.5 Self-check — what SRS guarantees

4.6 Stratified sampling

4.7 Combining strata: the weighted estimator

Suppose stratum \(h\) has size \(N_h\) and its sample gives proportion \(\hat{p}_h\) in favour. The fair population estimate weights each stratum by its true size:

The next three slides are one running example (the "Adelaide 500" survey) that pins down exactly when pooling works and when it fails.

4.8 Worked example I — equal strata, equal samples

20,000 voters: 10,000 City + 10,000 North Adelaide. SRS of 250 from each. In favour: 60/250 City (0.24), 190/250 North (0.76).

4.9 Worked example II — equal strata, unequal samples

Same 10,000 + 10,000 strata. Now SRS of 400 City + 100 North. In favour: 100/400 City (0.25), 75/100 North (0.75).

4.10 Worked example III — unequal strata AND unequal samples

Strata are now 12,000 City + 8,000 North. SRS of 400 City + 100 North. In favour: 100/400 (0.25), 75/100 (0.75).

4.11 Fairness vs precision

Two separate questions about an estimate:

5.1 Binary data and the sample proportion

Many discrete variables are binary — two outcomes:

• good / defective component
• agree / disagree with a statement
• approve / disapprove of a policy

5.2 Frequency distribution with table()

A corner shop orders 20 baguettes a day. Over 10 days the numbers unsold were: 0, 0, 1, 0, 1, 0, 0, 0, 2, 0.

x = c(0, 0, 1, 0, 1, 0, 0, 0, 2, 0)
T = table(x)                              # tally each distinct value
T.df = as.data.frame(T)
colnames(T.df) <- c("unsold", "frequency")
Unsold    = as.numeric(as.character(T.df$unsold))
Frequency = T.df$frequency

Note the as.numeric(as.character(...)) step: table() stores the values as factor labels, so we convert them back to numbers before plotting.

5.3 Needle (line) chart

# empty plot ("n" = draw nothing yet)
plot(Unsold, Frequency, type = "n",
     xlim = range(Unsold),
     ylim = c(0, max(Frequency)),
     xlab = "Unsold baguettes",
     ylab = "Frequency",
     xaxt = "n")            # no default x-axis

# draw the vertical needles
segments(x0 = Unsold, y0 = 0,
         x1 = Unsold, y1 = Frequency, lwd = 4)

# integer x-axis ticks
axis(1, at = Unsold)

Needles suit discrete data: each distinct value gets one spike, with nothing implied in between.

5.4 Needle chart with relative frequencies

Unsold = c(0, 1, 2)
RF     = c(0.7, 0.2, 0.1)

plot(Unsold, RF, type = "n",
     xlim = range(Unsold),
     ylim = c(0, max(RF)),
     xlab = "Unsold baguettes",
     ylab = "Relative frequency",
     xaxt = "n")

segments(x0 = Unsold, y0 = 0,
         x1 = Unsold, y1 = RF, lwd = 4)

axis(1, at = Unsold)

5.5 The same data as a bar chart

barplot(RF,
        names.arg = Unsold,
        width = 0.1, space = 5,
        col = "steelblue",
        border = "black",
        xlab = "Unsold baguettes",
        ylab = "Relative frequency")

Needle vs bar is a stylistic choice — both carry the same information for discrete data.

5.6 Applied task — loss of separation

Suggested approach

1. Read the spreadsheet — e.g. library(readxl); los <- read_excel("LOS.xlsx").
2. Build a frequency table of the incident counts with table(), as in 5.2.
3. Reuse the needle-chart recipe from 5.3: plot(..., type = "n"), then segments(), then axis().

This applies the whole Section 5 pipeline — tabulate, then draw — to a real dataset.

6.1 Synthesis

• We usually cannot calculate \(\mu\) — we don't know the variable for every member of the population.
• So we compute the sample mean \(\bar{x}\) as an estimate of \(\mu\).
• If we sample through simple random sampling — or a probability scheme with the right weights (stratification) — that estimate is fair.
• The accuracy of the estimate increases as the sample size grows.