Z scores

Presented by
Samarakoon Bandara

 Marks awarded by counting the number of
correct answers on a test script are known
as raw marks.
 Chandralatha got 16 marks out of 20 for
Psychology at the monthly test.
 Is it a high mark or a low mark? What do
you think?

 It may appear a high mark, but in fact the
statement is virtually meaningless on its
own.
 16 may be the lowest mark of a particular
group of scores. If the test was difficult, 16
may well be a very high mark.

Raw scores by themselves are not
very useful.
Raw scores need to be converted
into standard scores.

 A z-score (a standard score) indicates how
many standard deviations an element is from
the mean. A z-score can be calculated from
the following formula.
 z = (X - M) / SD
 where z is the z-score, X is the value of the
element, M is the population mean, and SD is
the standard deviation.

 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

 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 is no max/min
value.
 The z-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 standard deviations
above the mean", which means the latter
score is three times further from the mean.

 The Mean for a set of scores is 50 while the
SD is 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 SD) 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: The student has scored above
average - a distance of 1.96 above the average
score.

 A z-score less than 0 represents an element
less than the mean.
 A z-score greater than 0 represents an
element greater than the mean.
 A z-score equal to 0 represents an element
equal to the mean.

 A z-score equal to 1 represents an element
that is 1 standard deviation greater than the
mean; a z-score equal to 2, 2 standard
deviations greater than the mean; etc.
 A z-score equal to -1 represents an element
that is 1 standard deviation less than the
mean; a z-score equal to -2, 2 standard
deviations less than the mean; etc.

 If the number of elements in the set is large,
about 68% of the elements have a z-score
between -1 and 1; about 95% have a z-score
between -2 and 2; and about 99% have a z-
score between -3 and 3.

Z scores

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