Kapitel 04: Statistics for psychological assessments

Coaley, K. (2009): An Introduction to Psychological Assessment and Psychometrics. Sage.

STATISTICS FOR PSYCHOLOGICAL MEASUREMENT

• Outline the graphical representation of frequency distributions of test scores and the properties of the normal distribution.
Pictogram, histogram, frequency polygon. The normal distribution is symmetrical and has the mean, mode and median in the same place.

• How do you determine the measures of central tendency, incl. the standard deviation of distributions?
The mean is the average of all scores. The median is the middle score. The mode is the most common/frequent score. The variance is amount of variation between scores. The standard deviation indicates the extent to which scores are spread out.

• Explain how sampling affects error around mean scores and how we can use confidence limits to estimate this error.

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Outline the different types of standard scores, ex z scores, T scores, sten and stanine scores, how they are calculated and how they can be converted.
z scores: mean of 0, SD of 1, ranges from -3 to 3.
T scores: mean of 50, SD of 10, ranges from 20-80.
Sten scores: mean of 5,5, SD of 2, ranges from 1-10.
Stanine scores: mean of 5, SD of 2, ranges from 1-9.

FREQUENCY DISTRIBUTIONS

Can be graphically illustrated by: pictogram, histogram, frequency polygon etc.

THE MODE: the most common/frequently occuring score
o ex (1,2,2,5,6,7,9) mode = 2

THE MEAN: the average score
o ex (1,2,2,5,6,7,9) mean = 4,57

THE MEDIAN: the middle score
o ex (1,2,2,5,6,7,9) median = 5

THE NORMAL CURVE

• Aka bell curve aka normal frequency distribution 
aka Gaussian distribution.
• Is symmetrical and has the mean, the mode and the median in the same place.

μ = the mean, σ 2 is the variance (σ = the standard deviation)

Skewed distributions

• Can be due to a sampling bias, ex too many good performers or too easy items in a test.
• Reflects amount of asymmetry in a distribution.

MEASURES OF CENTRAL TENDENCY

• Standard deviation (SD) measures dispersion (spredning) of a sample. Indicates the extent to which scores are spread out.

…Hvis sample er meget lille divideres med (n-1)…

• Variance = a measure of the amount of variation between scores within a sample. The bigger the variance, the more scores differ from each other.

THE NORMAL CURVE AND PROBABILITY

Converting a frequency distribution into a probability distribution:

• Divide each frequency (ie. number of people at each point/score) by the total number of people in the sample.

SAMPLING AND STANDARD ERROR OF THE MEAN

• Sampling error of the mean = differences of mean values due to differences of the sample sizes.

The standard error of the mean

• SEmean = the SD of the distributions of means (of different samples with relation to the same measure). A frequency distribution of means of different samples.

CONFIDENCE LIMITS

Ex a frequency distribution of sample means. The probability that the ”true mean” will lie within 2 SEmeans is 68% (see above: The normal curve and probability), ie. there is a 68% confidence that the ”true mean” will lie within this range…
• A very large confidence range (ex 2 SEmeans on each side of the mean/95%) is very unclear and indicates that there can’t be much certainty of where the true mean lies.

THE NORMAL CURVE AND STANDARD SCORES

Standard scores involve measurement on an interval scale (unlike percentiles which represent an ordinal scale) and their norms represent scores having means and standard deviations chosen for their usefulness.

z scores
The most basic standard score, which can also be used to calculate other standard scores with.
• Mean = 0
• SD = 1

Converting raw scores to z scores:
• Subtract the mean from the score
• Divide the result by the standard deviation

ex: raw score = 36, mean = 25, SD = 8
1) 36 – 25 = 11
2) 11/8 = 1,375 = the z score

Normalizing scores

Normalized scores provide a common standard scale for different tests.
• Percentiles can be converted to z scores using the normal curve table, ex:
• When calculating z scores from percentiles rather than from raw scores, the resulting scores will be normally distributed even if the test’s raw scores are not.

T scores
• Mean = 50
• SD = 10
• Usually ranges from 20-80

Sten scores
• Mean = 5,5
• SD = 2
• Ranges from 1-10

Stanine
• Mean = 5
• SD = 2
• Ranges from 1-9

Converting z scores to other scales:
Score on new scale = Z score times the standard deviation of the new scale plus the mean of the new scale.

NEWscore = (Zscore x SDnew) + MEANnew eller 

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