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Glacius
StatisticsConcept reference

Mean intervals

A one-sample t interval adds and subtracts a t critical value times the estimated standard error from the sample mean under stated sampling conditions.

2 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. A sample mean gives one estimate of a population mean
  3. For data 4,4,4,84,4,4,8, the mean is 55
  4. The arithmetic alone does not validate the procedure
  5. The interval estimates the population mean
  6. Key takeaway
  7. Sources & further reading
  8. Concept connections

01A sample mean gives one estimate of a population mean#

A sample mean gives one estimate of a population mean. A one-sample t interval adds a margin reflecting uncertainty from having only a sample and estimating the population spread. Under the IID normal model, the t critical value uses n−1n-1 degrees of freedom.

For an IID normal sample with unknown population standard deviation, a one-sample t interval estimates the population mean. Use at least two observations and the sample SD based on denominator n−1n-1.

xˉ±t∗sn\bar x\pm t_*\frac{s}{\sqrt n}

Center at sample mean xˉ\bar x. Multiply estimated standard error s/ns/\sqrt n by the supplied critical value t∗t_*. Its degrees of freedom are n−1n-1.

For sample size 9, sample mean 15, and sample SD 6, first calculate SE=6/9=2SE=6/\sqrt9=2. With supplied critical value 2.306, the margin is 2.306(2)=4.6122.306(2)=4.612. Subtract and add that margin to the mean to obtain [10.388,19.612][10.388,19.612]. The margin is not the standard deviation of individual observations.

For n=9n=9, mean 1515 and sample SD 66, SE is 22. A supplied 95% critical value 2.3062.306 gives margin 4.6124.612.

Under the stated IID normal model, sample size 9, sample mean 15 and sample SD 6 give SE 6/sqrt(9)=2. With supplied 95% critical t=2.306, margin 2.306 times 2=4.612. The interval [10.388,19.612] is centered on 15; equal horizontal distances represent equal numeric distances.Under the stated IID normal model, sample size 9, sample mean 15 and sample SD 6 give SE 6/sqrt(9)=2. With supplied 95% critical t=2.306, margin 2.306 times 2=4.612. The interval [10.388,19.612] is centered on 15; equal horizontal distances represent equal numeric distances.
Figure 1Under the stated IID normal model, sample size 9, sample mean 15 and sample SD 6 give SE 6/sqrt(9)=2. With supplied 95% critical t=2.306, margin 2.306 times 2=4.612. The interval [10.388,19.612] is centered on 15; equal horizontal distances represent equal numeric distances.
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Check your reasoning

IID normal sample: n=4, mean=6, sample SD=2. For 95%, use t*=3.182. Rounded interval?

  1. A[2.818, 9.182]
  2. B[-0.364, 12.364]
  3. C[4.409, 7.591]
Show answer and explanation
[2.818, 9.182]

SE=1; margin=3.182.

02For data 4,4,4,84,4,4,8, the mean is 55#

For data 4,4,4,84,4,4,8, the mean is 55. Squared deviations add to 1212. Dividing by n−1=3n-1=3 gives variance 44, so sample SD is 22.

For 4,4,4,84,4,4,8, mean 55 and sample SD 22 give SE 11. Using the supplied 95% critical value 3.1823.182, the interval is [1.818,8.182][1.818,8.182]. Critical values here are rounded. Round interval endpoints to three decimals.

Check your reasoning

IID normal sample: 3, 3, 3, 7. Sample variance divides by n-1. For 95%, use t*=3.182. Rounded interval?

  1. A[1.244, 6.756]
  2. B[0.818, 7.182]
  3. C[-2.364, 10.364]
Show answer and explanation
[0.818, 7.182]

SE=1; margin=3.182.

03The arithmetic alone does not validate the procedure#

The arithmetic alone does not validate the procedure. Normality and independent sampling are separate conditions. A small estimated standard error cannot repair dependent sampling. Coverage refers to repeated intervals under the stated model.

Check your reasoning

IID normal sample: n=16, mean=3, sample SD=8. For 95%, use t*=2.131. “SE=8.” Repair rounded interval.

  1. A[-14.048, 20.048]
  2. B[1.935, 4.066]
  3. C[-1.262, 7.262]
Show answer and explanation
[-1.262, 7.262]

SE=2; margin=4.262.

04The interval estimates the population mean#

The interval estimates the population mean. It need not contain 95% of individual data values. The stated coverage depends on the sampling and distribution assumptions; correct arithmetic cannot compensate for systematically biased or dependent sampling.

Key takeaway

For the stated IID normal model, compute s/sqrt(n), multiply by the supplied t critical value, then subtract and add the margin around the sample mean.

  • Construct both t-interval endpoints.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Mean intervals.” Math behind ML. /learn/s-mean-interval