PSYB06H3 Chap.2 Distributions, Sampling and Standard Error
Distributions, Sampling and Standard Error
Shape comes before the summary
A data set gathered in large numbers settles into a recognisable form. The normal curve is symmetric with its mean, median and mode at one point, a skewed set stretches a tail toward its extreme scores, a uniform set spreads evenly, and some sets have no clear form at all.
Knowing the form tells you whether the mean is a fair description and whether normal-curve probabilities will apply later in the course.
Extreme values and what they change
A single score far from the rest can shift the mean noticeably while the median barely moves. This is the practical face of the trade-off between using every score and resisting unusual ones.
Screening for impossible values, such as a count larger than the number of items presented, belongs at the start of every analysis.
Why samples understate spread
Sample means are unbiased: they scatter evenly around μ. Sample variances are not, because a handful of draws tends to cluster near the centre of the population and miss its tails.
Dividing the sum of squares by n − 1 offsets that tendency, and the correction matters most for small samples.
From one sample to a statement about μ
If you could draw thousands of samples of the same size, their means would form a sampling distribution that becomes normal and narrower as n grows. The standard error, σ divided by the square root of n, gives the spread of that distribution without any resampling.
Adding and subtracting one standard error from a sample mean gives the range in which the population mean is expected to lie, for example the mean a whole population would reach if everyone received a treatment.
Precision has a price
Because the sample size sits under a square root, gains in precision slow down as samples grow.
Moving from 9 to 36 participants halves the standard error, but halving it again needs 144. Researchers therefore choose a target precision before collecting data, and exam questions often ask for exactly that calculation in reverse: given a standard deviation and a desired standard error, find n. Squaring the ratio of the standard deviation to the target error answers it, rounding any fractional sample upward.
The same logic explains why two studies with similar means can deserve very different confidence.
What this chapter covers
- 01
Normal, Skewed and Uniform Distributions
- 02
Kurtosis and Outliers
- 03
Unbiased Means and Biased Variances
- 04
Degrees of Freedom
- 05
Sampling Distributions and the Central Limit Theorem
- 06
Standard Error and Expected Ranges
Worked example · free
An expected range for a trained population
- 1Square root of the sample size: √36 = 6.
- 1Standard error: σ / √n = 18 / 6 = 3.
- 1Expected range: 104 ± 3, so from 101 to 107.
Key terms
- Normal Distribution
- A symmetric, bell-shaped probability distribution whose mean, median and mode share one value.
- Positive Skew
- A distribution shape in which the extreme scores lie above the mean, producing a long right tail.
- Kurtosis
- The degree to which a roughly normal distribution concentrates more or fewer scores near its mean than the normal curve does.
- Outlier
- A score lying an extreme distance from the mean, capable of shifting non-resistant statistics such as the mean.
- Sampling Distribution
- The distribution of a statistic, such as the mean, across many samples of the same size from one population.
- Grand Mean
- The mean of all scores across conditions, which must weight each condition mean by its sample size when groups differ in size.
Distributions, Sampling and Standard Error FAQ
Why does the standard error fall as the sample grows?
The standard error divides σ by the square root of n, so larger samples produce means that cluster more tightly around μ. Intuitively, a mean built from many scores lets high and low values offset each other. Because of the square root, quadrupling the sample is needed to halve the error.
Is the standard error the same as the standard deviation?
No. The standard deviation describes how individual scores vary around their mean. The standard error describes how sample means would vary around the population mean across repeated samples of the same size, so it is always smaller than σ whenever n is greater than one. Report the one that matches the question.
Does the central limit theorem require normal raw data?
No. Its point is that the distribution of sample means moves toward a normal shape as n increases even when the population itself is skewed or uniform. Larger samples bring the means closer to normal, which is what later z and t tests rely on when they treat a sample mean as normal.
How do I combine means from groups of different sizes?
Multiply each group mean by its own n to recover the group totals, add those totals, and divide by the combined N. Simply averaging the group means gives every group equal weight, which is correct only when all the groups are exactly the same size.
Exam move
Draw the three common distribution shapes from memory and mark where the mean and median fall on each. Then work a standard error problem in both directions: from σ and n to an expected range, and from a target error back to the sample size required.
Finish with a grand mean for unequal groups so the weighting step becomes routine, and check each answer by asking whether it moved in the direction a larger sample should move it.
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