54 - z score explained

7/28/2019 54 - Z Score Explained

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Z-scores

Another useful transformation in statistics is standardization. Sometimes called

"converting to Z-scores" or "taking Z-scores" it has the effect of transforming the originaldistribution to one in which the mean becomes zero and the standard deviation becomes

1. A Z-score quantifies the original score in terms of the number of standard deviationsthat that score is from the mean of the distribution. The formula for converting from an

original or "raw" score to a Z-score is:

The following data will be used as an example.

id REASON CREATIVE ZCREATIV

1 15 12 -.23

2 10 13 .04

3 7 9 -1.05

4 18 18 1.41

5 5 7 -1.60

6 10 9 -1.05

7 7 14 .31

8 17 16 .86

9 15 10 -.78

10 9 12 -.23


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11 8 7 -1.60

12 15 13 .04

13 11 14 .31

14 17 19 1.68

15 8 10 -.78

16 11 16 .86

17 12 12 -.23

18 13 16 .86

19 18 19 1.68

20 7 11 -.51

Figure 4.8 Data for creativity and logical reasoning example.

The mean for Creativity is 12.85 and the sd = 3.66

Therefore the z-score for case #5 (a raw creativity score of 7) =

The z-score for case #7 (a raw creativity score of 14) =


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A negative Z-score means that the original score was below the mean. A positive Z-score

means that the original score was above the mean. The actual value corresponds to the

number of standard deviations the score is from the mean in that direction. In the firstexample, a raw creativity score of 7 becomes a z-score of -1.60. This implies that the

original score of 7 was 1.6 sd units below the mean. In the second example, a z-score of

0.31 implies that the raw score of 14 was 0.31 standard deviations above the mean.

The process of converting or transforming scores on a variable to Z-scores is calledstandardization. There are other things in psychological science called "standardization",

so if someone says they "standardized" something, that doesnt necessarily mean they

converted raw scores to standard scores. A distribution in standard form has a mean of 0and a standard deviation of 1. However, it is important to note that a z-score

transformation changes the central location of the distribution and the average variability

of the distribution. It does not change the skewness or kurtosis.

Comparing Scores From Different Distributions. When scores are transformed to a Z-

score, it is possible to use these new transformed scores to compare scores from differentdistributions. Suppose, for example, you took an introductory research methods unit and

your friend studied English. You got a 76 (of 100) and your friend got 82 (also of 100).Intuitively, it might seem that your friend did better than you. But what if the class he

took was easier than yours? Or what if students in his class varied less or more than

students in your class in terms of final marks? In such situations, it is difficult to comparethe scores. But if we knew the mean and standard deviations of the two distributions, we

could compare these scores by comparing their Z-scores.

Suppose that the mean mark in your class was 54 and the standard deviation was 20 and

the mean mark in your friends class 72 and the standard deviation was 15. Your Z score

is (76-54)/20 = 1.1. Your friends Z score is (82-72)/15 = 0.67. So, using standard scores,you did better than your friend because your mark was more standard deviations above

the class mean than your friends was above his own class mean.

In this example, the distributions were different (different means and standard deviations)but the unit of measurement was the same (% of 100). Using standard scores or

percentiles, it is also possible to compare scores from different distributions where

measurement was based on a different scale. For example, we could compare two scoresfrom two different intelligence tests, even if the intelligence test scores were expressed in

different units (eg, one as an intelligence quotient and one as a percent of answers given

correctly). All we would need to know are the means and standard deviations of the

corresponding distributions.

Alternatively, we could have used other measures of relative standing to compare across

distributions. Scores on standardized tests are often expressed in terms of percentiles, for

example. This way, you can know how you did in comparison to other people who tookthat test that year, in the last five years, or whatever the time period being used when the

percentiles were constructed. For example, if your score was in the 65th percentile and

your friends was in the 40th percentile, you could justifiably claim that you did better


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than your friend because you outperformed a greater percent of students in your class

than he did in his class.

Another approach to understand Z score

Z-score explained

The z-score is associated with the normal distribution and it is a number that may be used

to:

tell you where a score lies compared with the rest of the data, above/below mean.

compare scores from different normal distributions

Let's take a closer look at each of these uses.

Z-score and distance from the mean, and Z-table

The Z-score is a number that may be calculated for each data point in a set of data.The

number is continuous and may be negative or positive, and there s no max/min value. Thez-score tells us how "far" that data point is from the mean. This distance from the mean is

measured in terms of standard deviation. We may make statements such as "the data

point(score) is 1 standard deviation above the mean" and "the score is 3 standarddeviations above the mean", which means the latter score is three times further from the

mean.

Example. The score for each student of a quiz is entered into a spreadsheet and a

histogram created from the data. Suppose the histogram is bell-shaped, symmetrical

about mean of 50 and st. dev of 10. What can we say about someone with a score of

50? The z-score is (50-50)/10 = 0. Interpretation: student score is 0 distance (in

units of standrd deviations) from the mean, so the student has scored average.

60? The z-score is (60-50)/10 = 1. Interpretation: student has scored above

average - a distance of 1 standard deviation above the mean.

69.6? The z-score is (69.6-50)/10 = 1.96. Interpretation: student has scored above

average - a distance of 1.96 above the average score.

Now, you might say after these examples that the z-score hasn't told you anything youcan't see without doing any calculations. But the z-score can tell you a bit more. Because

not only can it say whether a score is above or below the mean by so many st. dev., but itcan tell you what proportion of the data is below or above a particular score. In otherwords you can make statements such as "so and so got X marks in a quiz. The

corresponding Z-score is Y, and we can say that only Z per cent of students scored

higher". And this is where the z-table comes in.

Looking back the final example. The z-score was 1.96. Now if we look in the z-table we

find that 2.5 per cent of the scores is above 1.96. So we can say that in the population this


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student did better than 97.5 per cent of students, or that only 2.5 per cent of students

scored higher. What about for the student with z-score of zero - what proportion of

students did better?

Comparing scores from different normal distributions using the z-score

Suppose a student sits 2 exams, getting 55 in a verbal test and 60 in a numericalreasoning test. The class scores for each exam are normally distributed. For the verbal

test, the mean is 50 and standard deviation 5; for the numerical test, the mean is 50 and

standard deviation is 12.

Now it is plain to see that the student did above average for each test, and did better atnumerical reasoning. How did this student perform relative to everyone else? We can

answer this by calculating the z-score.

The z-score for the verbal test is (55-50)/5 = 1.

The z-score for the numerical test is (60-50)/12 = 0.83

Since the z-score for the verbal test is larger than for the numerical test, the student didbetter in the verbal than in the numerical test compared to everyone else. Another way to

see this is that z-score of 1 for verbal implies about 16 per cent did better at verbal; a z-

score of 0.83 for numerical implies about 20 per cent did better at numerical.

54 - z score explained

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