PSY2041 Chap.4 Descriptive Statistics
Descriptive Statistics
Descriptive Statistics begins with the variable and its measurement scale, then uses distribution shape, centre and spread to summarise observations honestly. Nominal, ordinal, interval and ratio scales preserve different information. Skew and outliers change how means, medians and variability should be interpreted.
Standard scores express distance from a mean in standard-deviation units, while correlation standardises covariance to describe linear association. Each statistic needs an interpretation in the original context.
What this chapter covers
- 01
Nominal, ordinal, interval and ratio measurement scales
- 02
Frequency distributions, symmetry, positive skew and negative skew
- 03
Mean, median and mode as different summaries of centre
- 04
Range, interquartile range, variance and standard deviation
- 05
Standard scores and distance from the reference mean
- 06
Covariance and Pearson correlation as linear association
- 07
Outliers, restricted range and causation boundaries
Worked example · free
Compute and interpret two standard scores
- 1Noor's deviation is 62 minus 50 = 12.
- 1Noor's z score is 12 divided by 8 = 1.50, so the score is 1.5 standard deviations above the mean.
- 1Eli's deviation is 42 minus 50 = -8.
- 1Eli's z score is -8 divided by 8 = -1.00, so the score is one standard deviation below the mean.
- 1Their standardised positions differ by 2.50 standard deviations in this reference distribution.
Key terms
- Measurement scale
- The information preserved by values, from categories and order to equal intervals and meaningful zero.
- Positive skew
- A distribution with a long right tail that commonly pulls the mean above the median.
- Interquartile range
- The spread of the middle half of ordered observations.
- Standard deviation
- A measure of typical score dispersion around the mean in the variable's original unit.
- Standard score
- A score expressed as distance from a reference mean in standard-deviation units.
- Pearson correlation
- A standardised coefficient describing the direction and strength of a linear association.
Descriptive Statistics FAQ
When should I use the median instead of the mean?
The median is often more representative when a quantitative distribution is strongly skewed or contains extreme observations. The mean remains informative but is pulled toward the tail. Report the shape and a suitable spread measure rather than choosing by habit.
What does a negative standard score mean?
It places the score below the reference mean. The magnitude gives the distance in standard-deviation units. It does not mean a negative amount of the construct, and its interpretation depends on the direction and meaning of the original scale.
Does a strong correlation prove causation?
No. Correlation describes how two measured variables vary together, usually in a linear pattern for Pearson's coefficient. Reverse direction, third variables, selection and measurement problems can explain the association. The research design determines whether causal language is defensible.
Exam move
Start every problem by naming the measurement scale and distribution shape. Build a small decision table linking scale and shape with appropriate centre and spread. Compute z scores with the subtraction order printed, then interpret the sign before the magnitude. For skew, sketch the tail and show which way it pulls the mean.
When writing about correlation, include variables, direction, approximate strength and linear form, followed by a causation boundary. Use tiny invented datasets to recompute means, medians and ranges by hand, because a visible outlier makes the contrast memorable. Always return the statistic to the original variable and reference group.
Build four miniature datasets, one for each measurement scale, and justify which operations preserve meaning. Recode the nominal categories with different numbers to prove that their mean changes arbitrarily, then order ordinal responses and identify what information their ranks retain. For quantitative observations, calculate the mean, median, range, interquartile range and standard deviation before adding one extreme value.
Record which summaries move and connect that movement with the distribution's tail. Practise z scores in three forms: a value above the mean, one below it and one exactly at the centre. Write the formula before substitution, keep the negative sign and translate the result into standard-deviation units. Do not call a z score a percentage or percentile.
For correlation, sketch a positive linear cloud, a negative cloud, a curve and a pattern dominated by one point. Predict what a single coefficient will reveal or conceal in each case. Add a subgroup colour and ask whether the combined trend matches the within-group trends. Complete every exercise with a sentence naming the sample, variables and reference frame.
Then add one limitation concerning outliers, restricted range, nonlinearity or causal inference. This workflow makes descriptive choices visible and stops arithmetic from replacing interpretation. Keep an error log for sign reversals, scale mismatches and causal wording.
Recalculate each example without notes a day later, then compare the numerical step and the interpretation separately so a correct computation cannot conceal a conceptual mistake.
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