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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 4,4,4,8, the mean is 5
  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 4,4,4,8, the mean is 5#

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