# How to compute a confidence interval for a single population variance

## Description

Let’s say we want to compute a confidence interval for the variability of a population. We take a sample of data, $x_1, x_2, x_3, \ldots, x_k$ and compute its variance, $s^2$. How do we construct a confidence interval for the population variance $\sigma^2$?

## Using SciPy, in Python

View this solution alone.

We’ll use R’s dataset EuStockMarkets here. This dataset has information on the daily closing prices of 4 European stock indices. We’re going to look at the variability of Germany’s DAX closing prices. For more information on how this is done, see how to quickly load some sample data. You can replace the example data below with your actual data.

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from rdatasets import data
import pandas as pd
df = data('EuStockMarkets')

# Select the column for Germany's DAX closing prices
sample = df['DAX']


Now that we have our sample data loaded, let’s go ahead and make the confidence interval using SciPy.

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from scipy import stats

# Find the critical values from the right and left tails of the Chi-square distribution
alpha = 0.05       # replace with your chosen alpha (here, a 95% confidence level)
n = len( sample )
left_critical_val = stats.chi2.ppf(1-alpha/2, df=n-1)
right_critical_val = stats.chi2.ppf(alpha/2, df=n-1)

# Find the upper and lower bounds of the confidence interval and print them out
lower_bound = ((n - 1)*sample.var())/left_critical_val
upper_bound = ((n - 1)*sample.var())/right_critical_val
lower_bound, upper_bound

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(1104642.2801539514, 1256248.1273200295)


Our 95% confidence interval for the standard deviation of Germany’s DAX closing prices is $[1104642, 1256248]$.

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## Solution, in R

View this solution alone.

We’ll use R’s dataset EuStockMarkets here. This dataset has information on the daily closing prices of 4 European stock indices. We’re going to look at the variability of Germany’s DAX closing prices. Feel free to replace this sample data with your actual data if you use this code.

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# install.packages("datasets") # if you have not done so already
library(datasets)

# Load stock market data, convert to DataFrame, and choose the DAX column.
EuStockMarkets <- data.frame(EuStockMarkets)
sample <- EuStockMarkets$DAX  Now that we have our sample data loaded, let’s go ahead and make the confidence interval. 1 2 3 4 5 6 7 8 9 10 11 # Find the critical values from the right and left tails of the Chi-square distribution alpha <- 0.05 # replace with your chosen alpha (here, a 95% confidence level) n <- length(sample) left_critical_val <- qchisq(p = alpha/2, df = n-1, lower.tail=FALSE) right_critical_val <- qchisq(p = 1-alpha/2, df = n-1, lower.tail=FALSE) # Find the upper and lower bounds of the confidence interval and print them out lower_bound <- ((n - 1)*var(sample))/left_critical_val upper_bound <- ((n - 1)*var(sample))/right_critical_val lower_bound upper_bound  1 2 3 4 5 [1] 1104642 [1] 1256248  Our 95% confidence interval for the standard deviation of Germany’s DAX closing prices is$[1104642, 1256248]\$.