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

It is calculated as:

Confidence Interval = x  +/-  t*(s/√n)

where:

• x: sample mean
• t: t-value that corresponds to the confidence level
• s: sample standard deviation
• n: sample size

This tutorial explains how to calculate confidence intervals in Python.

Confidence Intervals Using the t Distribution

If we’re working with a small sample (n <30), we can use the from the scipy.stats library to calculate a confidence interval for a population mean.

The following example shows how to calculate a confidence interval for the true population mean height (in inches) of a certain species of plant, using a sample of 15 plants:

import numpy as np
import scipy.stats as st

#define sample data
data = [12, 12, 13, 13, 15, 16, 17, 22, 23, 25, 26, 27, 28, 28, 29]

#create 95% confidence interval for population mean weight
st.t.interval(alpha=0.95, df=len(data)-1, loc=np.mean(data), scale=st.sem(data)) 

(16.758, 24.042)

The 95% confidence interval for the true population mean height is (16.758, 24.042).

You’ll notice that the larger the confidence level, the wider the confidence interval. For example, here’s how to calculate a 99% C.I. for the exact same data:

#create 99% confidence interval for same sample
st.t.interval(alpha=0.99, df=len(data)-1, loc=np.mean(data), scale=st.sem(data)) 

(15.348, 25.455)

The 99% confidence interval for the true population mean height is (15.348, 25.455). Notice that this interval is wider than the previous 95% confidence interval.

Confidence Intervals Using the Normal Distribution

If we’re working with larger samples (n≥30), we can assume that the sampling distribution of the sample mean is normally distributed (thanks to the ) and can instead use the from the scipy.stats library.

The following example shows how to calculate a confidence interval for the true population mean height (in inches) of a certain species of plant, using a sample of 50 plants:

import numpy as np
import scipy.stats as st

#define sample data
np.random.seed(0)
data = np.random.randint(10, 30, 50)

#create 95% confidence interval for population mean weight
st.norm.interval(alpha=0.95, loc=np.mean(data), scale=st.sem(data))

(17.40, 21.08)

The 95% confidence interval for the true population mean height is (17.40, 21.08).

And similar to the t distribution, larger confidence levels lead to wider confidence intervals. For example, here’s how to calculate a 99% C.I. for the exact same data:

#create 99% confidence interval for same sample
st.norm.interval(alpha=0.99, loc=np.mean(data), scale=st.sem(data))

(16.82, 21.66)

The 95% confidence interval for the true population mean height is (17.82, 21.66).

How to Interpret Confidence Intervals

Suppose our 95% confidence interval for the true population mean height of a species of plant is:

95% confidence interval = (16.758, 24.042)

The way to interpret this confidence interval is as follows:

There is a 95% chance that the confidence interval of [16.758, 24.042] contains the true population mean height of plants.

Another way of saying the same thing is that there is only a 5% chance that the true population mean lies outside of the 95% confidence interval. That is, there’s only a 5% chance that the true population mean height of plants is less than 16.758 inches or greater than 24.042 inches.