Samples, Error Bars & Confidence
A sample is a small window on a whole population. Learn why samples wobble, how error bars show that wobble, and when two groups really differ.
You almost never measure everyone. A drug trial tests a few hundred patients, not the whole country; a lab counts a few dozen cells, not every cell in the dish. From that small sample you have to guess the truth about the whole population, and every sample wobbles a little just by chance (that wobble is sampling error). The whole point of statistics is to say how much you should trust the guess. Bigger, truly random samples wobble less, so they give tighter estimates. The standard error (SE) puts a number on the wobble of a sample mean, error bars draw that wobble on a graph, and a confidence interval (CI) states the plausible range for the real value. Epidemiologists use these ideas to say whether a disease rate really rose or just jumped around by chance. Biostatisticians build the confidence intervals that decide if a new drug beat the placebo. Lab scientists check whether two error bars overlap before claiming a treatment changed anything. Doctors reading a study look at the CI to see if a result is strong or shaky before changing how they treat patients. Learn this and you stop being fooled by a difference that is really just noise.
- Common Core · HSS-IC.B.4Use data from a sample survey to estimate a population mean, and understand that different samples give different estimates (this spread is the margin of error the sample carries).
- Common Core · HSS-IC.B.6Evaluate reports based on data: judge whether a claimed difference between two groups is supported once the uncertainty in each estimate is taken into account.
- Common Core · HSS-ID.A.4Use the mean and standard deviation of a data set, and the normal model, to estimate what fraction of values fall in a given range (the basis of the 68-95-99.7 rule).
- Ohio · Ohio HS S.IC.4Estimate a population value from a random sample and describe how much the estimate could vary from one sample to the next.
- NGSS · SEP-4Analyzing and Interpreting Data: consider the limitations of a data set, including sample size and measurement error, when deciding what the data actually show.
- Find the mean (average) of a data set: A sample mean is the estimate everything here is built on, so students must be able to compute an average first.
- Read standard deviation as spread: Standard error is standard deviation divided by the square root of the sample size, so students need what standard deviation measures.
- Take a square root of a whole number: The SE formula divides by the square root of n, so students must evaluate roots like the square root of 36 or 225.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
Put a number on the wobble. The standard error (SE) equals the standard deviation divided by the square root of the sample size, and it tells you how precise the sample mean is. Draw it as error bars, and use it to build a confidence interval: mean give or take about two standard errors is a 95% range for the true value.
A sample of 36 patients has a standard deviation of 12 beats per minute. What is the standard error of the mean? (SE = standard deviation divided by the square root of n)
Reviewed- A.0.33 beats per minute
- B.2 beats per minute
- C.6 beats per minute
- D.72 beats per minute
Show the worked solution ▾
Answer: B. 2 beats per minute
- Step 1: Find the square root of n: The sample size n is 36, and the square root of 36 is 6.
- Step 2: Divide: SE = 12 divided by 6 = 2 beats per minute.
Why it's right: SE = 12 divided by the square root of 36 = 12 divided by 6 = 2 beats per minute.
- A: This divides 12 by 36 instead of by the square root of 36.
- C: This is the square root of n itself, not the standard error.
- D: This multiplies 12 by 6 instead of dividing.
Aligned to Ohio HS S.IC.4: compute standard error · reading level ~grade 9
The graph compares two experiments. In each experiment, which pair of groups shows a difference you can more confidently call real?
Reviewed- A.Groups A and B, because their error bars overlap
- B.Groups C and D, because their error bars are separated
- C.Both pairs equally, because all bars are drawn
- D.Neither pair, because you cannot tell from error bars
Show the worked solution ▾
Answer: B. Groups C and D, because their error bars are separated
- Step 1: Look at A and B: Their error bars overlap, so the true values could be the same; the gap between the bar tops might just be noise.
- Step 2: Look at C and D: Their error bars do not overlap; there is a clear gap. That makes it more likely the two groups truly differ, not just by chance.
Why it's right: Clearly separated error bars (C and D) point to a real difference, while overlapping bars (A and B) mean the difference might just be sampling noise.
- A: Overlapping bars weaken, not strengthen, a claim of a real difference.
- C: Being drawn does not make a difference real; overlap is what matters.
- D: Error bars are exactly the tool used to judge whether a difference is real.
Aligned to Common Core HSS-IC.B.6: judge a difference from uncertainty · reading level ~grade 9
A sample mean is 50 beats per minute with a standard error of 2. Using mean give or take about two standard errors, what is an approximate 95% confidence interval for the true mean?
Reviewed- A.49 to 51
- B.48 to 52
- C.46 to 54
- D.42 to 58
Show the worked solution ▾
Answer: C. 46 to 54
- Step 1: Find two standard errors: Two times the standard error is 2 times 2, which is 4.
- Step 2: Add and subtract from the mean: 50 minus 4 is 46, and 50 plus 4 is 54, so the interval is 46 to 54.
Why it's right: A 95% CI is about mean plus or minus two SE: 50 minus 4 to 50 plus 4 gives 46 to 54.
- A: This uses one half of an SE, not two full SE.
- B: This uses only one SE (plus or minus 2), which is roughly a 68% interval, not 95%.
- D: This uses four SE (plus or minus 8), which is wider than the 95% interval.
Aligned to Ohio HS S.IC.4: build a confidence interval · reading level ~grade 9
- A student computes SE from a lab's standard deviation and sample size, then draws matching error bars on the class graph.
- A lab group reports a result as 50 plus or minus 4 (a 46 to 54 confidence interval) instead of a bare number.
- A reader sees two treatment bars whose error bars overlap and decides the study has not proven a difference.
Fill these in as you work through the lesson.
- Sampling error (the natural wobble between samples):
- Standard error (SE) (standard deviation over the square root of n):
- Confidence interval (CI) (mean plus or minus about two SE):
- Overlapping error bars (difference might just be noise):
The standard error equals the standard deviation divided by the square root of ; a 95% confidence interval is about the mean plus or minus standard errors; if two groups' error bars , the difference might just be noise.
- In your own words, why do two random samples from the same population give slightly different means?
- A sample has standard deviation 12 and n = 36. Show that SE = 2, then give the 95% confidence interval if the mean is 50.
- Two bars have overlapping error bars. What can and cannot you conclude about the difference between the groups?
A sample of 36 has a standard deviation of 12 and a mean of 50. First SE = 12 divided by the square root of 36 = 12 divided by 6 = ____. Then a 95% confidence interval is 50 plus or minus two SE, which is 50 plus or minus ____ , giving ____ to ____.
The vocabulary of this topic, shown in the way you will meet it.