PSYB06H3 Chap.5 The Normal Curve and z Scores
The Normal Curve and z Scores
Probability as area
For continuous data, a probability is the fraction of the area under a distribution that a region covers. The normal curve is symmetric with its mean, median and mode at the same point, so half the area lies on each side of the mean.
Because a single value has no width, questions always concern ranges: above a score, below it, or between two scores.
Standardising with z
A z score expresses how many standard deviations a value lies from its mean, so z equals the score minus the mean, divided by the standard deviation.
Standardised data centre on zero with unit standard deviation, which is why one printed table serves every normally distributed variable. The same idea allows a fair comparison of scores from different classes or even different kinds of measurement, such as speed and height.
A raw score that looks higher can still be the weaker result once each class spread is taken into account, because a large standard deviation makes the same distance from the mean less unusual.
Three table columns
The course table gives, for each positive z, the area from the mean to z, the larger portion and the smaller portion.
For a score below the mean the smaller portion is the lower tail, so a sign error turns a correct lookup into the wrong answer. Ranges on one side of the mean subtract mean-to-z areas; ranges that straddle the mean add them.
Reverse questions read the table from the body outward, then solve X equals the mean plus z times the standard deviation.
From binomial bars to a smooth curve
With many events, adding exact binomial probabilities by hand becomes impractical. The binomial mean NP and standard deviation equal to the square root of NPQ turn a count into a z score, and the table then gives an approximate ranged probability.
The approximation tightens as the number of events grows because the discrete bars fill in the curve. Proportions convert to head counts by multiplying by group size, rounding the number of people up. Written answers earn more credit when they show the z score, the column read and the final area together, since each step can then be checked independently by the reader.
What this chapter covers
- 01
Area Under the Normal Curve
- 02
Standardising Scores With z
- 03
Mean to z, Larger and Smaller Portions
- 04
Ranges Between Two Scores
- 05
Finding a Score From an Area
- 06
Normal Approximation to the Binomial
Worked example · free
How many people respond faster than one participant
- 1Standardise: z = (351 − 420) / 60 = −69 / 60 = −1.15.
- 1Faster means a smaller time, the lower tail beyond z = −1.15: smaller portion .1251.
- 1Slower covers the rest of the curve, the larger portion: .8749.
- 1Between 351 ms and the mean is the mean-to-z area: .3749.
Key terms
- Normal Distribution
- A symmetric bell-shaped distribution in which the mean, median and mode coincide.
- Standardisation
- Rescaling a score by subtracting the mean and dividing by the standard deviation to produce a z score.
- Mean to z
- The area under the normal curve between the mean and a given z score.
- Smaller Portion
- The area beyond a z score on the side away from the mean, the tail of the distribution.
- Larger Portion
- The area on the mean's side of a z score, equal to the mean-to-z area plus one half.
- Normal Approximation
- Using a z score built from NP and the square root of NPQ to estimate a binomial range probability.
The Normal Curve and z Scores FAQ
Why can I not find the probability of one exact score on the normal curve?
A single point on a continuous scale has no width, so the area above it is zero and the table has nothing to report. Questions are therefore framed as ranges: above a score, below it or between two scores. A binomial count is different because each discrete outcome has a bar with width.
How do I know whether to add or subtract two table areas?
Sketch the curve and shade the region. If both scores lie above the mean, or both lie below it, their mean-to-z areas overlap, so subtract the smaller from the larger. If the scores fall on opposite sides of the mean, the two areas do not overlap, so add them to get the range between the scores.
What changes when a z score is negative?
The table prints only positive values, and the curve is symmetric, so the areas for a negative z match those for its positive twin. What changes is location: the smaller portion becomes the lower tail and the larger portion covers everything above the score. Keep the sign when solving for a raw score.
When should I use the z approximation for a binomial question?
Use it when the number of events makes adding exact probabilities tedious, roughly beyond twenty events. Compute the mean as NP and the standard deviation as the square root of NPQ, standardise the count, and read the table. Expect the answer to differ slightly from the exact sum, less so as N grows.
Exam move
Drill table reading until each of the three columns can be found without hesitation, then add sign changes. For every practice question, sketch the curve and shade the requested region before touching the table, and write the z score on the sketch. Alternate forward questions, from a score to an area, with reverse questions, from an area to a score.
Close each session with one comparison across two distributions and one binomial question solved by the normal approximation.
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