A margin of error doesn't shrink in proportion to your sample size — it shrinks with the square root of it, so doubling your respondents narrows the interval by only about 30%, and it takes four times the sample to actually cut the margin in half. Drag the slider below and watch the exact numbers move.
Set the slider above to 100 and the margin reads ±9.8 points, so the interval runs from 40.2% to 59.8%. Double the sample to 200 and the margin only falls to ±6.9 — a real improvement, but nowhere near half. Quadruple the original sample to 400 and the margin drops to ±4.9, which is exactly half of ±9.8.
That is the whole rule. Margin of error is proportional to 1 divided by the square root of n, so multiplying n by 4 multiplies the margin by 1/2, and multiplying n by 2 only multiplies it by roughly 0.71. A survey that wants a tighter margin is not buying precision at a fixed price — every extra point of precision costs more responses than the last one did.
Three points on the slider make the pattern concrete. At 95% confidence:
Going from 100 to 1,000 is a 10x increase in sample size for roughly a 3x reduction in margin, not a 10x one. That gap between how much data you add and how much precision you get back is the entire reason survey sample sizes look strange at first glance — nobody uses 10,000 respondents for a general-interest poll, because the margin below a few hundred more responses stops moving in any way a reader would notice.
The calculator holds the underlying proportion at 50/50 on purpose, and does not let you type in a different one. That is the assumption that produces the largest possible margin for any given sample size — p(1 − p) is at its maximum exactly when p is 0.5 — so it is the standard worst-case number pollsters quote when they do not yet know the real split.
If the true proportion is far from 50/50 — say a poll where 90% of respondents agree — the actual margin at the same sample size would be smaller than what this tool shows, because p(1 − p) falls as p moves away from 0.5. Holding it fixed here keeps the two levers this tool is actually about, sample size and confidence, from being tangled up with a third variable.
At a sample of 1,000, switching the confidence level from 95% to 99% widens the margin from ±3.1 points to ±4.1 points — with the exact same 1,000 responses. Nothing about the data changed; the interval only has to be wide enough to be right more often, 99 times out of 100 instead of 95, and that extra certainty is bought with width rather than with more responses.
This is the trap in a question like "would increasing the confidence level increase the margin of error?" — it is tempting to think confidence and precision move together, but they trade against each other at a fixed sample size. More confidence, wider interval, same survey.
The Digital SAT does not ask you to compute a margin of error from a formula — there is no z-score to look up on the reference sheet. It asks you to reason about a margin you are given: if a poll reports 62% support with a margin of error of 3 percentage points, which range of values is plausible for the true population proportion, and would a stated claim (like "more than 60% support this") be consistent with that interval.
The two facts this tool makes visible are exactly the two the test leans on: a larger sample produces a narrower interval, and a higher confidence level produces a wider one at the same sample size. A question that describes a survey being repeated with more respondents, or with a different confidence level, is asking you to move one of these two sliders in your head and say which direction the margin goes.
Drag the sample-size slider to see the margin, the interval and its width update together. The two radio buttons switch the confidence level between 95% and 99% at whatever sample size the slider is set to, so you can isolate that lever from the sample-size one.
The lower bar on the chart always shows four times the sample size in the slider, with its own margin, so the exact halving relationship is visible side by side rather than something you have to trust. The readout beneath the controls spells out the arithmetic — the z-score, the square root, and the resulting margin — for whatever position the slider is in.
Interactive Percent Change Calculator: Why 20% Up and 20% Down Isn't 0%
See why a 20% rise followed by a 20% fall doesn't get you back to where you started, and why stacking two 50%-off coupons is 75% off, not free. Drag two percent changes and watch the exact combined multiplier and final amount update in real time.
What does a practice SAT score range actually mean?
A score range is the platform admitting how much it does not know. The centre is its best estimate of your scaled score; the width is how far that estimate could be off given the evidence behind it. A single number from a practice test carries the same uncertainty — it just hides it.
How should you read your practice test results?
Read the domain breakdown before the score. The total tells you where you are; the breakdown tells you what to do on Monday. A 690 with one weak domain is about a week of work — a 690 that is evenly weak is closer to a month.
Every interactive tool: the full set. How a practice result turns into a score range: the scoring method.
SAT® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this site. All practice questions and reading passages are original works created by Preptics and are not actual SAT® questions.
Margin of error: why precision gets expensive
Notice that the bar does not shrink as fast as the sample grows. Doubling n only divides the margin by about 1.41; it takes four times the sample to halve it. And switching to 99% widens the interval without a single extra response.
The margin is z x the square root of p(1 - p) / n, so here 1.96 x the square root of 0.25 / 1,000 = 0.0310, or +/- 3.1 percentage points. Because n sits under a square root, halving the margin always costs four times the sample: 1,000 responses become 4,000 just to go from +/- 3.1 to +/- 1.5. Doubling to 2,000 gets you only to +/- 2.2.
Confidence is the other lever, and it is free of data: at 99% the same 1,000 responses give +/- 4.1 points instead of +/- 3.1. More confidence, wider interval, same survey.
