Statistics Confidence Intervals

A confidence interval gives a range of plausible values for a population parameter, built from sample data, along with a stated long-run success rate for the method used to build it.

What a Confidence Interval Really Means

A confidence interval is a range built from sample data that is meant to capture a population parameter, like a mean or a proportion, along with a stated confidence level such as 95%.

Note: That 95% describes the reliability of the method across many repeated samples, not the probability that this one interval contains the true value. The precise statement is: if you repeated the sampling and interval-building process many times, about 95% of the resulting intervals would contain the true population parameter. Any single interval either contains the true value or it doesn't - there's no probability left to assign once it's built.

Building a CI for a Mean (Sigma Known)

Example

import math

def z_confidence_interval(sample_mean, pop_std, n, z_crit=1.96):
    margin = z_crit * (pop_std / math.sqrt(n))
    return sample_mean - margin, sample_mean + margin

lower, upper = z_confidence_interval(sample_mean=72.5, pop_std=9.0, n=50)
print("95% CI:", (round(lower, 2), round(upper, 2)))

Building a CI for a Mean (Sigma Unknown)

When the population standard deviation isn't known - the far more common situation - we estimate it from the sample and use a critical value from the t-distribution with n - 1 degrees of freedom instead of the normal distribution. Because the t-distribution has heavier tails, this produces a slightly wider interval that accounts for the extra uncertainty of estimating the spread from the same data used to estimate the mean.

Example

import math

def sample_std(data):
    mean = sum(data) / len(data)
    variance = sum((x - mean) ** 2 for x in data) / (len(data) - 1)
    return math.sqrt(variance)

def t_confidence_interval(sample, t_crit):
    n = len(sample)
    mean = sum(sample) / n
    s = sample_std(sample)
    margin = t_crit * (s / math.sqrt(n))
    return mean - margin, mean + margin

# t critical value for 95% confidence, df = n - 1 = 14, looked up from a t-table
scores = [61, 64, 59, 70, 68, 62, 65, 60, 66, 63, 69, 58, 67, 64, 61]
lower, upper = t_confidence_interval(scores, t_crit=2.145)
print("95% CI:", (round(lower, 2), round(upper, 2)))
Confidence Levelz critical value (two-tailed)
90%1.645
95%1.960
99%2.576
  • A larger sample size shrinks the interval, since standard error falls as sample size grows
  • A higher confidence level (say 99% instead of 90%) widens the interval
  • More variability in the underlying data widens the interval
Note: Doubling your sample size does not halve the interval width - because standard error depends on the square root of n, you'd need to roughly quadruple your sample size to cut the margin of error in half.
  1. Decide on a confidence level, such as 95%
  2. Compute the sample mean and standard error
  3. Look up the appropriate critical value (z if sigma is known, t with n - 1 degrees of freedom otherwise)
  4. Multiply the critical value by the standard error to get the margin of error
  5. Add and subtract the margin of error from the sample mean to get the interval

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