Sampling distributions, standard error, and 95% continence intervals
Sampling distributions
Estimate: an inference about a population parameter based on a subsample of that population. Since samples are only a subset of the entire population, there is a chance that estimates from repeated sampling from that population can be off by a particular amount. We can visualize this uncertainty by plotting a sampling distribution.
For this figure, the true population mean is 43.92 mm.
Standard error: The standard deviation of the sampling distribution.
95% confidence interval: the range likely to contain the true population mean.
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| # Dealing with uncertainty in R | |
| library(palmerpenguins) | |
| data(penguins) | |
| penguins <- na.omit(penguins) | |
| # If we want to calculate 95% CI using the 2SE convention: | |
| species <- NULL | |
| means <- NULL | |
| std_devs <- NULL | |
| sizes <- NULL | |
| sems <- NULL | |
| cis <- NULL | |
| # Counter | |
| k <- 1 | |
| for (i in levels(penguins$species)) { | |
| x <- penguins$bill_length_mm[penguins$species == i] | |
| means[k] <- mean(x) | |
| std_devs[k] <- sd(x) | |
| sizes[k] <- length(x) | |
| sems[k] <- std_devs[k] / sqrt(sizes[k]) | |
| cis[k] <- paste(means[k] – 2 * sems[k], means[k] + 2 * sems[k], sep = ",") | |
| species[k] <- paste(i) | |
| k <- k + 1 | |
| } | |
| results <- data.frame(species, sizes, means, std_devs, sems, cis) | |
| names(results) <- c("species", "n", "mean", "sd", "se", "ci") | |
| print(results) | |
| # Mean, standard deviation, standard error, and 95% confidence intervals using **t-distribution** | |
| library(Rmisc) | |
| rmisc_results <- summarySE(penguins, measurevar = "bill_length_mm", groupvars = "species") | |
| ci_lower <- rmisc_results$bill_length_mm – rmisc_results$ci | |
| ci_upper <- rmisc_results$bill_length_mm + rmisc_results$ci | |
| print(paste(ci_lower, ci_upper, sep = ",")) |
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| # Dealing with uncertainty in Python | |
| import pandas as pd | |
| import numpy as np | |
| from scipy.stats import t | |
| from palmerpenguins import load_penguins | |
| penguins = load_penguins().dropna() | |
| # Split the data frame into groups | |
| groups = penguins.groupby('species') | |
| # Calculate sample means, standard deviations, and standard errors for each group | |
| means = groups['bill_length_mm'].mean() | |
| std_devs = groups['bill_length_mm'].std() | |
| sizes = groups['bill_length_mm'].size() | |
| sems = groups['bill_length_mm'].sem() | |
| # Calculate 95% CIs for each group | |
| cis = {} | |
| for group, mean, std_dev, size, sem in zip(means.index, means, std_devs, sizes, sems): | |
| ci_1 = mean – (2 * sem) | |
| ci_2 = mean + (2 * sem) | |
| cis[group] = ci_1, ci_2 | |
| print(cis) | |