BIOL2001 Chap.9 Standard Error, z-Scores and the Central Limit Theorem
Standard Error, z-Scores and the Central Limit Theorem
From samples to populations
Population quantities such as the mean, variance and standard deviation are parameters; the same quantities computed from a sample are statistics that estimate them.
The chapter explains why the sample variance divides by n minus 1, which makes it an unbiased estimator, and works the calculation for five snail masses.
z-scores and normal areas
A z-score counts how many standard deviations a value lies from the mean, turning any normal variable into the standard normal so one table of areas serves every case.
A table works through areas above, below and between values for invented wing lengths.
Sampling distributions
The standard error, sigma divided by the square root of n, measures how far a sample mean is likely to stray from the population mean.
A computed figure shows the sampling distribution of the mean narrowing as the sample grows, and the Central Limit Theorem explains why sample means are approximately normal even when the population is not.
What this chapter covers
- 01
Parameters versus statistics
- 02
Sample variance with the n minus 1 divisor
- 03
Standardising values with z-scores
- 04
Areas with pnorm and lower.tail
- 05
Standard error of the mean
- 06
Standard deviation versus standard error
- 07
The Central Limit Theorem
- 08
When the t distribution takes over
Worked example · free
How sample size shrinks the standard error
- 1SE = 0.24 / √4 = 0.12 g for n = 4.
- 1SE = 0.24 / √16 = 0.06 g and 0.24 / √64 = 0.03 g: each quadrupling of n halves the SE.
- 1For n = 16, z = (0.90 − 0.80) / 0.06 = 1.67.
- 1The area above z = 1.67 is about 0.048.
Key terms
- Population Parameter
- A fixed numerical property of a whole population, such as its mean or standard deviation.
- Unbiased Estimator
- A statistic whose average over all possible samples equals the parameter it estimates.
- Z-Score
- The number of standard deviations a value lies above or below the mean.
- Sampling Distribution
- The distribution of a statistic, such as the mean, over repeated random samples.
- Central Limit Theorem
- The result that sample means are approximately normal for large samples, whatever the population's shape.
Standard Error, z-Scores and the Central Limit Theorem FAQ
What is the difference between standard deviation and standard error?
The standard deviation describes how individual observations spread around the mean. The standard error describes the typical distance between a sample mean and the population mean, and equals the standard deviation divided by the square root of the sample size, so it shrinks as samples grow.
Why does the sample variance divide by n minus 1?
Dividing the sum of squared deviations by n minus 1 makes the sample variance an unbiased estimator of the population variance. Dividing by n would underestimate the population's spread on average, which matters most for small samples.
What does the Central Limit Theorem say?
It says that, for a large enough sample, the distribution of sample means is approximately normal with mean mu and variance sigma squared over n, even when the population itself is not normal. That is why normal areas can be used for questions about means.
Assessment move
Work the snail example by hand and then with mean(), var() and sd() in R. For every area question, sketch the curve, mark the boundaries and shade the region before calculating any z-score. Practise three questions about single values and three about sample means side by side, so the switch from standard deviation to standard error becomes automatic.
As a final check of understanding, explain in two sentences why quadrupling a sample halves its standard error, and why that makes larger samples more likely to detect small effects.
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