psychology 2910 - lecture 9 (standard scores/z-scores)

What does the score "x = 75" tell you?

Not much:

- Better if the mean = 70

- Worse if the mean = 80

Also depends on standard deviation:

- Better if mean = 70 and SD = 1.0

- Not quite as good if mean = 70 and SD = 15.0

Why are percentiles useful?

Indicates where a score falls within a distribution

- Measure of relative standing

- Tells us how a score compares to the other scores (75% of scores are at or below this value)

Metric

The metric of a variable is how we understand what the numbers mean (e.g. measuring height with inches vs. centimetres)

Often with interval measures, the metric is defined in terms of the mean and SD, and is often arbitrary (e.g. IQ)

We can rescale the metric by doing a linear transformation

Linear Transformation

When you multiply, divide, add, or subtract a constant from each score in a distribution, the mean and/or standard deviation can change

- Assume mean on exam is 50 and SD is 20

- I may want to change this to mean = 70 and SD = 10

- Changing the metric like this will not affect any inferential statistics

If you add/subtract a constant to each score in a distribution, what will happen?

- The mean of that distribution will change by that constant

- The standard deviation will not change (i.e. if you shift the mean left or right, the shape of graph and spread of data will not change)

If you multiply/divide a constant from each score in a distribution, what will happen?

- The mean of that distribution will change by the same multiple

- The standard deviation of that distribution will change by the same multiple (i.e. because it stretches or compresses the spread of data)

Standard Score

A number in a set where the mean and standard deviation of the set are already known (i.e. a pre-defined metric)

Z-Score

- The classic standard score

- Mean = 0

- Standard deviation = 1

- Standardizing = converting to z-scores

- Tell you how many standard deviations each score is away from the mean

What is the definitional formula for a z-score?

Subtract the mean from the individual raw score and divide by the standard deviation

Suppose 40 people were administered a cognitive test where the mean score was 42 and the standard deviation was 3. Find the z-scores associated with the following 4 participants:

Mean = 42

SD = 3

n = 40

First z-score: 45 - 42 / 3 = 1

Second z-score: 48 - 42 / 3 = 2

Third z-score: 39 - 42 / 3 = -1

Fourth z-score: 36 - 42 / 3 = -2

Characteristics of Z-Scores

- A z-score near zero indicates that the score is at or near the mean

- A z-score of -1 indicates that the score is 1 standard deviation below the mean

- A z-score greater than ± 3 is extremely large and should be examined to determine if it is an outlier

How do you convert a z-score into a raw score?

Mean = 42

SD = 3

n = 40

- A participant has a z-score of 2.5

- What was the test score (x)?

Why is doing two transforms (i.e. converting a raw score to a z-score and vice versa) important?

Frequently do both:

- Raw score = z-score

- Z-score = another raw score

Why? Consider IQ:

- Convert raw score to z-score:

- Mean IQ = 0

- Very high IQ = 2.5

- Convert to mean = 100, standard deviation = 15

Who did better relative to other students (56 on Psychology vs. 60 on English exam)?

Need to know mean and standard deviation, then compute z-score:

- Psychology: mean = 50, SD = 5, x = 56

- English: mean = 54, SD = 10, x = 60

- Psychology: z = 56 - 50 / 5 = 1.2

- English: z = 60 - 54 / 10 = 0.6

- Psychology score is arguably better (i.e. when comparing standard deviations)

Assume a distribution of scores on an exam with a mean of 54 and a standard deviation of 14. However, I want a distribution with a mean of 70 and a standard deviation of 10.

What would a score of 64 equal on the old distribution? The new distribution?

Convert raw score to z-score, then convert z-score to raw score:

z = 64 - 54 / 14 = 0.71429

x = 70 + 0.71429 x 10 = 77.14

Therefore, the raw score of the new distribution would be 77.14

Standard vs. Normal

- Standardized = z-scores

- Normal Distribution = distribution has a particular bell shape to it

- Normal distribution is often called the z distribution, and z-scores are used to find points on the distribution

- However, converting something to z-scores does not make it normal

Standard Normal Distribution

- Approx. 66% of scores are between ± 1 SD of the mean; 68% is more accurate, but 2/3 is close enough

- Approx. 95% of scores are between ± 2 SD of the mean

- Approx. 99% of scores are between ± 3 SD of the mean

- Therefore, -3 < most z-scores < +3

How can we interpret a z-distribution table?

If z = 1.00:

- Area to the left = 0.84134

- 84.134% of the scores to the left of a z-score of 1.00

- 34.134% (84.134 - 50.00) of scores are between 0 (the mean) and 1 (the z-score)

- 15.866% of scores (1 - 0.84134) are greater than a z-score of 1.00

How would a z-distribution table be converted to a graph? How can you read this graph?

z = 1.00

Area from table = 0.84134 (84.134%)

Yellow area = 0.34134 (34.134%)

Blue area = 0.15866 (15.866%)

Negative Z-Scores

Exactly the same as positive z-scores except we are on the left side of the normal curve

If z = -1.00:

- Area to the left: 0.15866

- 34.134% (.50000 - .15866) of scores are between 0 (the mean) and -1 (the z-score)

- 15.866% of scores are less than -1.00

- 84.134% of scores (1 - .15866) are more than -1.00

How can we prove that approx. 66% of scores are within ± 1 SD (if data is distributed normally)?

If z = 1.00, Yellow Area = 0.3413

If z = -1.00, Yellow Area = 0.3413

0.3413 + 0.3413 = 0.6826 or 68.26%

How can we prove that approx. 95% of scores are within ± 2 SD (if data is distributed normally)?

If z = 2.00, Yellow Area = 0.4772

If z = -2.00, Yellow Area = 0.4772

0.4772 + 0.4772 = 0.9544 or 95.44%

How can we prove that approx. 99% of scores are within ± 3 SD (if data is distributed normally)?

If z = 3.00, Yellow Area = 0.4987

If z = -3.00, Yellow Area = 0.4987

0.4987 + 0.4987 + 0.9974 or 99.74%

What percentage of scores falls between the z-scores 1.00 and 1.50?

- Find area to left of z = 1.00

- Find area to left of z = 1.50

Subtract:

- If z = 1.00, area to left = 0.84134

- If z = 1.50, area to left = 0.93319

- 0.93319 - 0.84134 = 0.09185

Therefore: 9.185%