Would a Statewide Sample Pick My School?

Class topic: MATH& 146, Sections 1.3 and 1.4 — the frame, simple random sampling, and systematic, stratified and cluster sampling.

From first through eighth grade I went to Whidbey Island Waldorf School, a private school. For high school I switched to South Whidbey High School, the public high school on the south end of the island. I graduated in 2011.

That switch gave me a question. Suppose Washington wanted to estimate what share of its public school students are low-income, but could visit only 50 schools. Would South Whidbey High be among them? And how much would the answer depend on the way the state picked?

Across the water in Mukilteo, Kamiak High School enrolled 2,284 students in 2025–26. South Whidbey High enrolled 405. A survey that picks schools has to account for a difference like that: each school gets a place on the list, but each school does not hold the same number of students.

Where the numbers come from

Every fall, Washington’s Office of Superintendent of Public Instruction (OSPI) counts the students in every public school on the first business day of October. It publishes the counts on the state’s open-data site.¹ One of the counts is low-income students. In Washington that means students who qualify for free or reduced-price school meals.² Statewide, it’s close to half of all students.

In class, the list we draw from is called a frame. Here it begins with Washington’s public schools. My Waldorf school is private, so it never appears. However carefully I sample this list, I cannot pick a school that is missing from it. Later, I narrow the list again to schools with the information this analysis needs.

I also wanted to compare schools by place. The National Center for Education Statistics sorts them into City, Suburb, Town, and Rural groups.³ South Whidbey High is Rural: Fringe; Kamiak is Suburb: Large. “Fringe” means South Whidbey High is in a rural area close to an urban area or town, according to the federal distance rules.

What I did in R

A short script, prepare-data.R, downloads both lists straight from the source, so anyone can rerun it. Then I loaded them into R.

schools <- read.csv("ospi_enrollment_2025-26_schools.csv",
                    colClasses = c(districtcode = "character", schoolcode = "character"))
locale  <- read.csv("nces_locale_2024-25_wa.csv", colClasses = c(seasch = "character"))
nrow(schools)   # every public school on the state's list
head(schools[, c("county", "districtname", "schoolname", "all_students", "low_income")])

I matched the state’s schools to their federal place labels. The state list had 2,413 schools; 2,343 had everything I needed for this comparison. I left out 19 without a matching federal label, 46 with no students counted, and 5 whose low-income counts were hidden for privacy.

Among the students in those 2,343 schools, 47.4% were counted as low-income. That is the answer for my final list. In class, an answer for the whole group is called a parameter. Each sample below tries to estimate it. It is not a figure for every public or private school student in Washington.

# Give each school the federal city / suburb / town / rural label
schools$seasch <- paste(schools$districtcode, schools$schoolcode, sep = "-")
locale$area <- ifelse(locale$urban_centric_locale >= 11,
                      c("City", "Suburb", "Town", "Rural")[pmax(1, locale$urban_centric_locale %/% 10)], NA)
frame <- merge(schools, locale[, c("seasch", "area")], by = "seasch")   # schools with no label drop out

# Keep schools that have students and a visible low-income count
frame <- subset(frame, all_students > 0 & low_income + non_low_income == all_students)
frame <- frame[order(frame$districtcode, frame$schoolcode), ]           # the order OSPI lists them
N <- nrow(frame)
N

# The true answer for the whole list (the parameter)
true_pct <- 100 * sum(frame$low_income) / sum(frame$all_students)
round(true_pct, 1)

Then I split the list by area. This table surprised me.

by_area <- aggregate(cbind(all_students, low_income) ~ area, data = frame, FUN = sum)
by_area$schools <- as.vector(table(frame$area)[by_area$area])
by_area$pct_low_income <- round(100 * by_area$low_income / by_area$all_students, 1)
by_area[, c("area", "schools", "all_students", "pct_low_income")]

subset(frame, schoolname %in% c("South Whidbey High School", "Kamiak High School"),
       c(schoolname, area, all_students, low_income))

Rural schools make up 22% of the schools in my list but enroll 13% of its students. There are many rural schools, but they tend to be smaller. In this list, suburban schools have the lowest low-income share and town schools the highest.

South Whidbey High and Kamiak looked more alike than I expected: 145 of 405 students (36%) at South Whidbey High and 767 of 2,284 (34%) at Kamiak were counted as low-income. Both are below the 47.4% for my final list. I had expected the rural school to have the higher share. These two schools overturned that expectation, though they cannot tell me what every rural or suburban school is like.

Four ways to pick 50 schools

Say the state can only visit 50 schools. Here are the four ways from class to pick them. I set a seed first (set.seed(146), after the class number). A seed makes R’s random picks repeatable: anyone who runs my code gets the same 50 schools I did.

1. Simple random sample (SRS): every school’s name goes in a hat, and I draw 50. Every group of 50 schools is equally likely.
2. Systematic sample: divide the list size by 50 to get a step size: 2,343 ÷ 50 = 46.9, rounded down to k = 46, as the textbook does. Pick a random starting school between 1 and 46, then take every 46th school down the list. Before this draw, I sort the list by district code. Rounding down has one catch: the last 43 schools on the list can never be picked. That’s under 2% of the list, so it barely moves the results. A bigger study would use a method that covers every school.
3. Stratified sample: split the list into groups first (City, Suburb, Town, Rural), then draw a random sample from each group. Each group gets a share of the 50 that matches its share of schools: 16 city, 16 suburban, 7 town and 11 rural.
4. Cluster sample: pick whole districts at random and take every school in them. I picked 7 districts, which comes to about 50 schools on average.

The one-line test I’ll remember: stratified = some schools from every group; cluster = every school from some groups.

set.seed(146)   # the class number, so anyone who reruns this gets the same schools
n <- 50         # schools per sample

# The estimate: out of all students in the picked schools, what % are low-income?
pct <- function(s) 100 * sum(s$low_income) / sum(s$all_students)

# 1. Simple random sample: 50 names from a hat
draw_srs <- function() frame[sample(N, n), ]

# 2. Systematic (textbook method): k = list size / 50, rounded down;
#    random start from 1 to k, then every kth school
draw_systematic <- function(data = frame) {
  k <- nrow(data) %/% n              # 2,343 / 50 = 46.86, rounded down to 46
  start <- sample(k, 1)
  data[seq(start, by = k, length.out = n), ]
}

# 3. Stratified: a random sample from each area, sized to match its share of schools
sizes <- round(n * table(frame$area) / N)
draw_stratified <- function() {
  do.call(rbind, lapply(names(sizes), function(a) {
    rows <- which(frame$area == a)
    frame[rows[sample(length(rows), sizes[[a]])], ]
  }))
}

# 4. Cluster: 7 whole districts at random, every school in them
districts <- unique(frame$districtcode)
draw_cluster <- function() frame[frame$districtcode %in% sample(districts, 7), ]

one <- list(SRS = draw_srs(), Systematic = draw_systematic(),
            Stratified = draw_stratified(), Cluster = draw_cluster())
data.frame(schools        = sapply(one, nrow),
           rural_schools  = sapply(one, function(s) sum(s$area == "Rural")),
           pct_low_income = round(sapply(one, pct), 1))

The answer for my full list is 47.4%. In this one set of draws, the stratified sample came closest. The cluster sample was 13 percentage points high. One draw can favor a method by luck, so I repeated the draws before comparing them.

None of these four draws picked South Whidbey High. In a simple random draw of 50 from 2,343 schools, its chance of appearing is about 2%, or 1 in 47. That is the answer to my opening question for one draw from this list.

One sample can get lucky, so I drew 1,000

One draw can land close by luck. To see how much the answers moved, I had R repeat each method 1,000 times. For each method, the chart shows the middle 95% of the resulting estimates. A shorter bar means the guesses varied less in these runs.

reps <- 1000
sims <- data.frame(
  method   = rep(c("SRS", "Systematic", "Stratified", "Cluster"), each = reps),
  estimate = c(replicate(reps, pct(draw_srs())),
               replicate(reps, pct(draw_systematic())),
               replicate(reps, pct(draw_stratified())),
               replicate(reps, pct(draw_cluster()))))

# Middle 95% of the 1,000 estimates for each method
spread <- do.call(rbind, lapply(split(sims$estimate, sims$method), function(x)
  data.frame(low = unname(quantile(x, 0.025)), mid = median(x), high = unname(quantile(x, 0.975)))))
spread$method <- rownames(spread)
rownames(spread) <- NULL
spread[, c("low", "mid", "high")] <- round(spread[, c("low", "mid", "high")], 1)
spread$width  <- spread$high - spread$low
spread[order(spread$width), c("method", "low", "mid", "high", "width")]

# Average guess for each method, and how much a cluster sample's size swings
round(tapply(sims$estimate, sims$method, mean), 1)
cluster_sizes <- replicate(reps, nrow(draw_cluster()))
range(cluster_sizes)

library(ggplot2)
spread$method <- factor(spread$method, levels = spread$method[order(spread$width, decreasing = TRUE)])
ggplot(spread, aes(y = method)) +
  geom_errorbar(aes(xmin = low, xmax = high), orientation = "y", width = 0.25) +
  geom_point(aes(x = mid), size = 2.5) +
  geom_vline(xintercept = true_pct, linetype = "dashed") +
  labs(x = "% of students who are low-income, estimated from one sample",
       y = NULL,
       title = "Same list, four ways to pick: how far each can miss",
       subtitle = paste0("Bar = middle 95% of 1,000 samples. Dot = middle guess.\nDashed line = true value for all ",
                         format(N, big.mark = ","), " schools (", round(true_pct, 1), "%).")) +
  theme_minimal(base_size = 12) +
  theme(plot.title.position = "plot")

Cluster sampling produced the widest spread of estimates here. A wide spread does not mean every cluster draw missed; it means the guesses varied the most. Schools in the same district often serve similar neighborhoods, so 7 districts can give fewer different views than 50 separate schools. The size of a cluster sample also swung, from 9 to 222 schools across the 1,000 draws. Its average estimate was 49.0%, about 1.6 points above the 47.4% for my list.

Stratified sampling narrowed the spread a little (13.8 points, against 14.3 for simple random sampling). It makes sure every kind of place shows up. But in this list the four areas’ low-income shares are not far apart, 42% to 55%, so guaranteeing each area a share changes less than it would if the groups differed more.

Systematic sampling had the narrowest spread here (40.9% to 52.4%, 11.5 points). I sorted schools by district code before sampling. That order may help spread the picks across districts. When I shuffled the list, the systematic estimates spread out more, to 41.2% to 55.1% (13.9 points). The comparison suggests that list order matters here; it does not prove that district grouping is the only reason.

# Test: does systematic still win if the list is shuffled first?
shuffled <- frame[sample(N), ]
shuffled_sys <- replicate(reps, pct(draw_systematic(shuffled)))
round(quantile(shuffled_sys, c(0.025, 0.975)), 1)

How confident can we be? These bars show what happened when I repeatedly sampled from the same 2,343-school list. They describe the luck of the draw within that list. They do not measure the effect of leaving schools out, using an imperfect low-income measure, or changing the list in another school year.

The class words for what I found

Every method here begins with a random choice. A convenience sample would be different: I might visit only schools I could drive to from Langley. Repeating that trip would not give the other schools a chance to appear. I want to look at that problem next.

Why these definitions matter beyond my example

My four samples are a classroom exercise. Real funding programs use different data and rules. Still, the labels and income measures I used led me to a practical question: what happens when a program has to define “rural” or “low-income”?

• A rural label can affect grant eligibility. For one federal rural education grant, every school in a district generally needs an NCES rural code of 41, 42, or 43; a state rural designation can also qualify. A second grant also allows some town codes. Each program has other requirements, so South Whidbey High’s label alone does not tell me whether its district qualifies.⁴
• The meal-income cutoff does not adjust for local rent. For the 2025–26 school year, a family of four in the contiguous United States could qualify for reduced-price school meals with income up to $59,478; the free-meal cutoff was $41,795.⁵ Those national thresholds do not rise because housing costs more in Langley.
• Funding can depend partly on survey data, though not on my school sample. Washington uses free or reduced-price meal eligibility data in its Learning Assistance Program calculations. Federal Title I allocations use Census Bureau school-district poverty estimates that combine survey information with other data.⁶ These programs measure different things from my 50-school exercise, but each has to decide whose circumstances its numbers capture.

What I still want to know about South Whidbey

The national meal-income cutoff made me think about what a dollar buys on the island. If I were deciding how to help students, I would want to know what opportunities they can reach as well as how many meet an income cutoff. A ferry trip to a training program on the mainland might matter to a student as much as aid paid directly to a school.

South Whidbey High is smaller than it was: OSPI counted 514 students in 2014–15 and 405 in 2025–26.⁷ That change feels familiar to me. When I was about 8, 9, or 10, cars lined the streets at Halloween and I could go through ten bags of candy. Now I get about five knocks a night. Trick-or-treaters are not a population count, but that memory made me look for one. The Census Bureau estimated that about 58% of Langley city residents were 65 or older and about 6% were under 20 in 2019–23. Those estimates are for Langley, not the whole south end, and they have wide margins of error.⁸

At Waldorf, my grade often had 12 students or fewer. My classes at South Whidbey High felt much bigger, and differences among classmates stood out more to me. Today’s enrollment figures cannot tell me whether the students were different when I attended. They do remind me how much the group around me shaped what I noticed.

In one simple random draw, South Whidbey High had about a one-in-47 chance of appearing. Before I trusted a statewide estimate, I would ask which schools could be picked, how they were picked, and which students the final percentage represents.

Sources: OSPI Report Card Enrollment 2025-26 · OSPI FRL report, 2025 (pp. 10–11) · NCES Locale Classifications · Urban Institute Education Data Portal, CCD directory 2024-25 · U.S. Dept. of Education, REAP · USDA Income Eligibility Guidelines 2025-26 · OSPI Poverty Measures Q&A · Census SAIPE school district estimates · Census ACS S0101, Langley. Photos: hero photo by Davin Fladager-McCullough; others from Wikimedia Commons, credited above. Data files: ospi_enrollment_2025-26_schools.csv and nces_locale_2024-25_wa.csv, downloaded by prepare-data.R.