Download THE NORMAL DISTRIBUTION CURVE

THE NORMAL DISTRIBUTION CURVE
AND Z-SCORES
LEARNING OBJECTIVES
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At the end of lecture students must be able to know,
Properties of the normal curve.
Mean and standard deviation of the normal curve.
Area under the curve.
Calculating z-scores.
Probability.
CARL GAUSS
 The normal curve is often called the Gaussian distribution,
after Carl Friedrich Gauss, who discovered many of its
properties.
 Gauss, commonly viewed as one of the greatest
mathematicians of all time (if not the greatest), is honoured
by Germany on their 10 Deutschmark bill.
PROPERTIES OF THE NORMAL
DISTRIBUTION:
1. IT IS BELL-SHAPED AND ASYMPTOTIC AT THE
EXTREMES.
2. IT'S SYMMETRICAL AROUND THE
MEAN.
3. THE MEAN, MEDIAN AND MODE
ALL HAVE SAME VALUE.
4. IT CAN BE SPECIFIED
COMPLETELY, ONCE MEAN AND SD
ARE KNOWN.
5. THE AREA UNDER THE CURVE IS
DIRECTLY PROPORTIONAL TO THE
RELATIVE FREQUENCY OF
OBSERVATIONS.
e.g. here, 50% of scores fall below the mean, as
does 50% of the area under the curve.
RELATIONSHIP BETWEEN THE NORMAL
CURVE AND THE STANDARD DEVIATION:
frequency
All normal curves share this property: the SD cuts off a
constant proportion of the distribution of scores:-
68%
95%
99.7%
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mean
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Number of standard deviations either side of mean
About 68% of scores fall in the range of the mean plus and minus 1 SD;
95% in the range of the mean +/- 2 SDs;
99.7% in the range of the mean +/- 3 SDs.
e.g. IQ is normally distributed (mean = 100, SD = 15).
68% of people have IQs between 85 and 115 (100 +/- 15).
95% have IQs between 70 and 130 (100 +/- (2*15).
99.7% have IQs between 55 and 145 (100 +/- (3*15).
68%
85 (mean - 1 SD)
115 (mean + 1 SD)
We can tell a lot about a population just from knowing the mean; SD;
and that scores are normally distributed.
If we encounter someone with a particular score, we can assess how
they stand in relation to the rest of their group.
e.g. someone with an IQ of 145 is quite unusual (3 SDs above the
mean).
IQs of 3 SDs or above occur in only 0.15% of the population [ (10099.7) / 2 ].
Z-SCORES:
z-scores are "standard scores".
A z-score states the position of a raw score in relation to
the mean of the distribution, using the standard
deviation as the unit of measurement.
z 
raw score  mean
standard deviation
for a population :
z 
X  μ
σ
for a sample :
z 
X - X
s
RAW SCORE DISTRIBUTIONS:
A score, X, is expressed in the original units of measurement:
X = 236
X = 65
X  50 s  10
X  200 s  24
z = 1.5
X0 s  1
z-score distribution:
X is expressed in terms of its deviation from the mean (in SDs).
Z-SCORES TRANSFORM OUR ORIGINAL
SCORES INTO SCORES WITH A MEAN OF 0
AND AN SD OF 1.
Raw IQ scores (mean = 100, SD = 15):
55
70
85
100
115
130
145
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IQ as z-scores (mean = 0, SD = 1).
z for 100 = (100-100) / 15 = 0,
z for 115 = (115-100) / 15 = 1,
z for 70 = (70-100) / -2, etc.
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THANK YOU
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