When to Use s / sqrt(n) in Statistics

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In statistics, you will encounter the formula s/√n in different scenarios.

This formula is used to calculate the standard error of a sample mean.

In the formula, s represents the sample standard deviation and n represents the sample size.

This formula appears in the calculation of two statistical tests:

1. One sample t-test

2. Confidence interval for a population mean

The following examples show how to use s/√n in both of these scenarios.

Example 1: Using s / sqrt(n) in a One Sample t-test

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

We use the following formula to calculate the test statistic t:

t = (x – μ) / (s/√n)

where:

• x: sample mean
• μ₀: hypothesized population mean
• s: sample standard deviation
• n: sample size

For example, suppose we want to test whether or not the mean weight of turtles in some population is equal to 300 pounds.

We collect a simple random sample of turtles with the following information:

• Sample size n = 40
• Sample mean weight x = 300
• Sample standard deviation s = 18.5

We will perform the one sample t-test with the following hypotheses:

• H₀: μ = 310 (population mean is equal to 310 pounds)
• H_A: μ ≠ 310 (population mean is not equal to 310 pounds)

First, we’ll calculate the test statistic:

t = (x – μ) / (s/√n) = (300-310) / (18.5/√40) = -3.4187

According to the T Score to P Value Calculator, the p-value associated with t = -3.4817 and degrees of freedom = n-1 = 40-1 = 39 is 0.00149.

Since this p-value is less than 0.05, we reject the null hypothesis. We have sufficient evidence to say that the mean weight of this species of turtle is not equal to 310 pounds.

Example 2: Using s / sqrt(n) in a Confidence Interval for a Population Mean

A confidence interval for a population mean is a range of values that is likely to contain a population mean with a certain level of confidence.

We use the following formula to calculate a confidence interval for a mean:

Confidence Interval = x  +/-  t_{n-1, 1-α/2}*(s/√n)

where:

• x: sample mean
• t: the t-critical value
• s: sample standard deviation
• n: sample size

For example, suppose we want to calculate a confidence interval for the true mean weight of turtles in a certain population.

We collect a simple random sample of turtles with the following information:

• Sample size n = 40
• Sample mean weight x = 300
• Sample standard deviation s = 18.5

We can use the following formula to calculate a 95% confidence interval for the true population mean weight of turtles:

• 95% C.I. = x +/-  t_{n-1, 1-α/2}*(s/√n)
• 95% C.I. = 300 +/- (2.022691) * (18.5/√40)
• 95% C.I. = [294.083, 305.917]

The 95% confidence interval for the true population mean weight of turtles is between 294.083 pounds and 305.917 pounds.

Additional Resources

The following tutorials explain how to calculate a standard error of a mean in different software:

How to Calculate the Standard Error of the Mean in Excel
How to Calculate Standard Error of the Mean in R
How to Calculate the Standard Error of the Mean in Python