(a) Differentiate between primary and secondary mental abilities. How the general mental ability scales differ from IQ test. (b) Y = mX + C is an equation of straight line. Draw a graph showing relationship between X and Y and relate the equation to the slope and intercept on the graph.

(a) Primary and secondary mental abilities

These terms come from factor analysis, which groups mental test scores that tend to rise and fall together. Usage varies slightly between authors, but the standard distinction is as follows.

Primary mental abilities are relatively narrow, basic abilities. L. L. Thurstone (1938) identified seven: verbal comprehension, word fluency, numerical ability, spatial visualisation, associative memory, perceptual speed and reasoning.

Secondary mental abilities are broader abilities that emerge when the primaries, which are themselves correlated, are analysed again. These are also called second-order factors. The best-known are in the theory of Raymond Cattell and John Horn: fluid intelligence, the capacity to reason and solve new problems, and crystallised intelligence, knowledge and skills acquired through education and experience.

Key differences: primary abilities are specific and are measured directly by particular tests, such as a test of vocabulary. Secondary abilities are general and are inferred from patterns across several primary abilities. A primary ability explains performance on a narrow set of tasks, while a secondary ability explains performance across a wide set.

General mental ability scales and IQ tests

General mental ability (GMA) refers to the general factor g, first proposed by Charles Spearman. A GMA scale estimates this general capacity and is often a short test used by groups, for example in employee selection or competitive examinations. Its score is commonly reported as a raw or percentile score.

An IQ test, such as the Wechsler or Stanford-Binet, is usually a longer, individually administered battery with several subtests, and is used for clinical or educational assessment. It reports a standardised score that compares a person with others of the same age, with a mean of 100 and a standard deviation of 15. Historically, IQ was calculated as (mental age ÷ chronological age) × 100.

The two differ mainly in length, purpose, administration and method of reporting, not in what they fundamentally aim to measure. Their scores are closely correlated, as both draw heavily upon g.

(b) Y = mX + C

In this equation, X is the independent variable, Y is the dependent variable, m is the slope and C is the Y-intercept.

Slope (m): the steepness of the line, or the change in Y for each one-unit change in X. Slope = rise ÷ run = (Y₂ − Y₁) ÷ (X₂ − X₁). If m is positive, the line rises from left to right. If m is negative, it falls. If m is zero, it is horizontal.

Intercept (C): the point where the line crosses the Y-axis, that is, the value of Y when X = 0. The line passes through the point (0, C).

Worked example: take Y = 2X + 1, so that m = 2 and C = 1.

When X = 0, Y = 1. When X = 1, Y = 3. When X = 2, Y = 5. When X = 3, Y = 7.

To draw the graph: draw a horizontal X-axis and a vertical Y-axis, and mark equal units on both. Plot the points (0, 1), (1, 3), (2, 5) and (3, 7), and join them with a ruler to form a straight line. Then mark C = 1, where the line meets the Y-axis. To show the slope, draw a small right-angled triangle on the line: moving 1 unit to the right gives a rise of 2 units, so m = 2 ÷ 1 = 2.

Relation: changing C moves the line up or down without altering its steepness, while changing m alters its steepness and direction but leaves the Y-intercept fixed.