PSYB06H3 Chap.1 Variables, Measurement and Descriptive Statistics
Variables, Measurement and Descriptive Statistics
Know what kind of number you have
Every analysis starts by classifying the data. Nominal data name categories with no order, ordinal data rank categories, interval data have equal spacing without a true zero, and ratio data add a true zero. The classification limits the arithmetic: a mean of nominal codes is meaningless, and a statement that one score is twice another only works on a ratio scale.
Grouping a measurement into categories is always possible, but the reverse is not.
Parameters, statistics and study roles
Population parameters use Greek letters and describe the whole group of interest; sample statistics use Roman letters and describe the subset actually measured.
In a study, the independent variable is manipulated, the dependent variable is measured, and a confound is anything that changes alongside the manipulation and could explain the result.
Short-answer questions often hinge on naming these roles correctly before discussing any number.
Summaries of centre and spread
The mean uses every score and sits as close as possible to all of them, the median splits ranked scores in half, and the mode is the most frequent value. Spread is described by the range, the interquartile range, the variance and the standard deviation.
Variance averages squared deviations because raw deviations always sum to zero. For a sample, the sum of squares is divided by n − 1 to remove the bias that dividing by n would introduce.
Displays that support the numbers
Frequency tables, bar graphs and histograms organise raw scores before they are summarised.
Histograms suit continuous variables, relative frequency versions divide by N, and cumulative versions stack totals. Bin width and a truncated vertical axis can both change the impression a chart gives, so read the scale before reading the shape.
What this chapter covers
- 01
Nominal, Ordinal, Interval and Ratio Scales
- 02
Population Parameters and Sample Statistics
- 03
Independent, Dependent and Confounding Variables
- 04
Mean, Median and Mode
- 05
Variance, Standard Deviation and Sums of Squares
- 06
Histograms, Binning and Truncated Axes
Worked example · free
Population and sample variance for five quiz scores
- 1Mean: (7 + 9 + 6 + 10 + 8) / 5 = 40 / 5 = 8.
- 1Deviations are −1, 1, −2, 2 and 0, so SS = 1 + 1 + 4 + 4 + 0 = 10.
- 1Population view: σ² = 10 / 5 = 2.0000 and σ = 1.4142.
- 1Sample view: s² = 10 / 4 = 2.5000 and s = 1.5811.
Key terms
- Nominal Data
- Categories with no order or numeric relationship, such as home city, which can be counted but not averaged.
- Ratio Data
- Measurements with equal spacing and a true zero, so both differences and multiplicative comparisons are meaningful.
- Confound
- An alternative explanation that changes alongside the manipulated variable and is not accounted for by the design.
- Median
- The middle score of ranked data, found at position (N + 1) / 2 and averaged across the middle pair when N is even.
- Interquartile Range
- The range of the middle 50% of ranked scores, Q3 minus Q1, which resists extreme values better than the full range.
- Relative Frequency
- The frequency of a score divided by the total number of scores, often shown as a percentage on a histogram.
Variables, Measurement and Descriptive Statistics FAQ
Why is the mean called the least squared distance value?
The mean is the single point that minimises the sum of squared distances to every score, so it sits as close as possible to all of the data at once. That is why it treats every score equally, and also why a single extreme score can pull it further than it pulls the median.
When should the median be reported instead of the mean?
Report the median, often with the interquartile range, when a few highly deviant scores would drag the mean away from where most scores lie. Final grades are a typical case: a handful of very low marks can lower the mean while the bulk of the class sits in a narrower band.
Why does a bar graph have gaps when a histogram does not?
A bar graph displays categories that have no continuous scale between them, so the bars stand apart. A histogram's horizontal axis is a continuous measurement, so adjacent intervals meet and the bars touch. Choosing the wrong display suggests a scale the data do not have.
Does changing the bin width change the data?
No, the scores stay the same, but the picture can change a great deal. Narrow bins can look noisy, while very wide bins can merge distinct groups and hide peaks. When comparing histograms, check that they use the same intervals before comparing their shapes.
Exam move
Make a one-line rule for each scale of measurement and test it on variables from your own studies. Recompute a mean, median, variance and standard deviation by hand for a small data set twice, once as a population and once as a sample, until the denominator choice is automatic. Then sketch a histogram of the same data and decide which summary of centre best describes it.
Finish by writing one sentence naming the independent variable, dependent variable and a plausible confound for a study you read about.
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