# Population Mean – Confidence Interval

Calculators

Use this calculator to determine a confidence interval for your sample mean where you are estimating the mean of a population characteristic (e.g., the mean blood pressure of a group of patients).

The estimate is your ‘best guess’ of the unknown mean and the confidence interval indicates the reliability of this estimate.  The confidence interval provides you with a set of limits in which you expect the population mean to lie.

# Calculator

This is your estimated mean calculated using a sample of data collected from your population.

This is your estimated standard deviation calculated using a sample of data collected from your population.

Typical choices are 90%, 95%, or 99%

%

The confidence level indicates the probability that the confidence interval will contain the true population mean.

This is the size of the sample you have used to calculate your sample mean.

How many people are there in the population from which you are sampling? The sample size doesn't change much for populations larger than 100,000.

# Alternative Scenarios

With a sample mean of
(19.38 , 20.62)
(99.38 , 100.62)
(499.38 , 500.62)
With a sample standard deviation of
(69.69 , 70.31)
(66.92 , 73.08)
(63.83 , 76.17)
With a confidence level of % % %
(69.48 , 70.52)
(69.38 , 70.62)
(69.19 , 70.81)
With a sample size of
(68.04 , 71.96)
(69.13 , 70.87)
(69.81 , 70.19)
With a population size of
(69.64 , 70.36)
(69.45 , 70.55)
(69.41 , 70.59)

### Worked Example

A study aims to estimate the mean systolic blood pressure of British adults by randomly sampling and measuring the blood pressure of 100 adults from the population.  From their sample, they estimate the sample mean to be 70mmHg and the sample standard deviation to be 8mmHg.  Using this information the 95% confidence interval is calculated as between 68.43 and 71.57mmHg.  If the blood pressure of a further 900 adults were measured then this confidence interval would reduce to between 69.51 and 70.49mmHg (assuming the estimated mean and standard deviation remained the same).

### Formula

This calculator uses the following formula for the confidence interval, ci:

ci = μ ± Zα/2*(s/√n)*√FPC,

where:

FPC = (N-n)/(N-1),

Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), μ is the sample mean, s is the sample standard deviation, n is the sample size and N is the population size. Note that a Finite Population Correction (FPC) has been applied to the confidence interval formula.

### Discussion

Calculating a confidence interval provides you with an indication of how reliable your sample mean is (the wider the interval, the greater the uncertainty associated with your estimate).

By changing the four inputs (the sample mean, sample standard deviance, confidence level and sample size) in the Alternative Scenarios, you can see how each input is related to the confidence interval. The larger your sample size, the more certain you can be that the estimate reflects the population, so the narrower the confidence interval. However, the relationship is not linear, e.g., doubling the sample size does not halve the confidence interval.

### Definitions

#### Sample mean

The sample mean is your ‘best guess’ for what the true population mean is given your sample of data and is calcuated as:

μ = (1/n)* ∑ni=1xi,

where n is the sample size and x1,…,xn are the n sample observations.

#### Sample standard deviation

The sample standard deviation is calcuated as s=√σ2, where:

σ2 = (1/(n-1))* ∑ni=1(xi-μ)2,

μ is the sample mean, n is the sample size and x1,…,xn are the n sample observations.

#### Confidence level

The confidence level is the probability that the confidence interval contains the true population mean.  If the study was repeated and the range calculated each time, you would expect the true value to lie within these ranges on 95% of occasions.  The higher the confidence level the more certain you can be that the interval contains the true mean.

#### Sample size

This is the total number of samples randomly drawn from you population.  The larger the sample size, the more certain you can be that the estimate reflects the population.  Choosing a sample size is an important aspect when desiging your study or survey.  For some further information, see our blog post on The Importance and Effect of Sample Size and for guidance on how to choose your sample size for estimates of the population mean, see our sample size calculator.

#### Population size

This is the total number of distinct individuals in your population.  In this formula we use a finite population correction to account for sampling from populations that are small.  If your population is large, but you don’t know how large you can conservatively use 100,000.  The sample size doesn’t change much for populations larger than 100,000.