Statistics in Psychology

Score distribution interpretation

A raw score has little meaning until it is located within the distribution of scores from which it came. The most common way to do this is the standard score (z-score), which expresses how far a score lies from the mean in standard deviation units: SC = \frac{X - M}{SD}, where X is the raw score, M is the mean, and SD is the standard deviation. The sign shows whether the score is above or below the mean, and the magnitude shows how unusual it is. Wechsler-type IQ scales express scores this way, so an IQ is read as a position relative to the norm group.

A raw score can look ordinary under one distribution and exceptional under another, so comparisons must use standard scores rather than raw scores. For a score above the mean, a smaller mean or a smaller SD produces a larger standard score. The most favourable interpretation of a fixed score is therefore the distribution that gives the largest z. Checking this requires computing z for each distribution. Standard scores can also be recoded onto other scales. A stanine is a standardized score on a nine-point scale from 1 to 9, with 5 as the middle value.

Standard error of estimate is a different concept from the standard score. It is the root mean square deviation of predicted values from true values, that is, the standard error of the prediction errors.

Variance and standard deviation

Variance is the mean of the squared deviations from the mean. For a population, \sigma^2 = \frac{\sum (X_i - \mu)^2}{N}, and the standard deviation is its positive square root, \sigma = \sqrt{\sigma^2}. Because variance is expressed in squared units, the standard deviation is the usual index of spread in the original units of the scores. If the variance is 100, the standard deviation is \sqrt{100} = 10, and only the positive root is taken. Standard deviation is also described as the root mean square deviation of scores from their mean.

Spread is measured by dispersion statistics. The range (maximum minus minimum), the variance, and the standard deviation are all measures of dispersion. ANOVA is an inferential test that partitions variance to compare group means, and correlation measures association between two variables, so neither is a measure of dispersion.

For a complete set of scores, the population formula divides by N. For a sample used to estimate a population, the divisor becomes n-1. For example, for the scores 2, 4, 6, 8, the mean is 5, the squared deviations are 9, 1, 1, 9 with sum 20, the population variance is 20/4 = 5, and the standard deviation is \sqrt{5} \approx 2.24. When the question treats the data as the full set, the population divisor applies. The same procedure applies to small data sets: sum the squared deviations from the mean, divide by the number of scores, and take the square root.

Two sets of scores can have the same mean but different spreads. Only the standard deviation reveals that difference, so a mean comparison alone is not enough to describe the data.

Shape of score distributions

A distribution is described by its symmetry and its tails. In a negatively skewed (left-skewed) distribution, the tail extends toward low values, so few scores lie at the low end and most scores cluster at the high end. The mean is then pulled toward the tail and is smaller than the median. A positively skewed (right-skewed) distribution has the opposite pattern, with a tail toward high values.

The normal distribution is a symmetric bell curve whose mean, median, and mode coincide. The area within fixed multiples of the standard deviation is constant. Between the mean and \pm 1\sigma lies about 34.13% of scores on each side, so the interval \mu \pm \sigma contains about 68.27% of cases. The intervals \mu \pm 2\sigma and \mu \pm 3\sigma contain about 95.45% and 99.73%.

To find the proportion beyond a given score, sum the tabled areas between standard deviation bands beyond that point. For a score one SD above the mean, the bands from 1\sigma to 2\sigma (about 13.59%), from 2\sigma to 3\sigma (about 2.14%), and beyond 3\sigma (about 0.13%) add up to about 15.86% of the population scoring higher. This assumes the scores are normally distributed.

Type I and Type II errors

In hypothesis testing, a Type I error is rejecting a true null hypothesis, and its probability is \alpha, the level of significance. A Type II error is failing to reject a false null hypothesis, and its probability is \beta. The power of a test is 1 - \beta. The two errors are linked: lowering one usually raises the other.

A courtroom analogy helps. With the null hypothesis that the accused is innocent, convicting an innocent person is a Type I error (wrongly rejecting a true null hypothesis), while acquitting a guilty person is a Type II error (failing to reject a false null hypothesis).

The level of significance is set by the researcher in advance and is not fixed by statistical theory. Common choices are 0.05, 0.01, and 0.10. Rejecting the null hypothesis at \alpha = 0.05 still carries a 5% chance of a Type I error, which is small but never zero. Wrong rejection is therefore always possible, although the probability can be made low.