This document discusses key concepts related to normal distributions and z-scores. It provides examples of how to calculate z-scores based on a data set's mean and standard deviation. It also shows how to use z-score values and the normal distribution table to determine the probability that a random value will fall within a given range. The key points are that z-scores indicate how many standard deviations a value is from the mean, and the normal distribution allows converting between z-scores and probabilities.

1. PSYC1004
Introduction to quantitative methods in
psychology
Session 5
1

2. Number Mean Deviation
Squared
deviations
1 3.20 -2.2 4.84
3 3.20 -0.2 0.04
2 3.20 -1.2 1.44
4 3.20 0.8 0.64
6 3.20 2.8 7.84
1 3.20 -2.2 4.84
1 3.20 -2.2 4.84
3 3.20 -0.2 0.04
7 3.20 3.8 14.44
4 3.20 0.8 0.64
39.6
4.40 2.10
Standard deviation (example)
2
Sample variance and SD ( i.e., 39.6/9) (SQRT of 4.40)
(sum of squared deviations)
square root of 4.40

3. The z score of a data point X
M = the mean of the dataset
S = the SD of the dataset
z score
3
Math (80 - 82) / 6 = -0.33
Verbal (75 - 75) / 3 = 0.00
Science (70 - 60) / 5 = 2.00
Logic (77 - 70) / 7 = 1.00
Z score
(Caldwell)
The z score of a raw score in a data set is the distance of the
data point from the mean in standard deviation units
eg both above average —> use sd and z score to see which perform better
sd can’t be negative
Text
Text

4. z score
It can be used to locate a score in a distribution of data: It
informs whether
• the score is above or below the mean; and
• the score’s deviation from the mean is relatively large or
relatively small compared with the typical deviations in the
dataset.
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Example:
• If a given z score is negative, the raw score being represented is (above/
below/ at) the population mean?
• Which of the following z scores represents a raw score that is the most atypical
(i.e., farthest from the mean)?
(a) −3.10 (b) -0.82 (c) 0.47 (d) 2.20
below
a

5. Histogram
5
...
When scores are measured on a continuous variable, the
distribution of frequency (number of cases or units in each value
range) is commonly displayed by a histogram.
z core will be the same
as the mean and the sd are the same
but the percentile not the same
percentile and z score is depend on the distribution

6. Population distribution
The distribution of a continuous variable in a large/infinite population
is typically represented graphically using a continuous line
(probability distribution curve). The histogram for a sample of data
from the population can be taken as an approximation of the curve.
6
(Source: Field)
pobility density
the meaning of the curve
larger sample —> better approximation —> line join become more like smooth cure

7. Probability distribution
The probability that a continuous variable is between two specified
values is equal to the area under the distribution curve over that
interval.
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Source: Howell
Probability
or relative
frequency probability finding someone under the value = the area under the curve

8. Normal distribution
• It is a family of theoretical probability distributions precisely generated
by a formula
• The distributions of many variables are taken/assumed as (or being
close to) normal distributions
• A normal distribution is symmetrical about its mean and extends to
infinity and negative infinity
• A bell-shaped probability distribution curve does not necessarily
represent a normal distribution.
8
probability
mean and sd —> curve
all normal distribution are bell shaped but bell shaped not eual to normal
distribution

9. Normal distribution
9
The approximate areas under the standard normal distribution
curve that lie between z = 0 and several whole-number z scores
(Source: Hatcher)
The percentage of the area under a normal distribution curve is a function of z score.

10. Normal distribution
10
Example: The probability of a
standardized and normally distributed
variable being less than 0.5,
P (z < 0.5) = .6915
.6915
P( -ꝏ < z < .5)

11. Normal distribution
P (z < 0.84) = 0.8
• As the total area under a
standard normal distribution
curve = 1,
P (z > 0.84) = 1 – 0.8 = 0.2
• As a normal distribution curve
is symmetrical,
P (z < - 0.84) = 0.2
11
0.8
.84
0.2
-.84
0.2

12. Example
What is the probability
that a standardized and
normally distributed
variable is between the
mean (z = 0) and 1.48?
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1.48
From the standard normal distribution table,
probability (z < 1.48) = .9306; i.e., referring to the diagram, the
area of A + B = 0.9306
Area of A = 0.5 (since a normal distribution is symmetrical about
its mean)
The required probability = the area of B = 0.9306 – 0.5 = 0.4306
B
A

13. Normal distribution
13
P(z < .8) = .7881
The probability of z being
between -1 and 0.8
P(z < -1) = . 1587
= .7881 - .1587 =.6294

14. Example
The mean height of the individuals in a population is 175 cm
and the standard deviation (SD) of their height values is 10
cm. Assuming that height is normally distributed in the
population and an individual is randomly selected, what is
the probability that the selected individual’s height is
between 165 and 185 cm?
In terms of z scores, the range of 165 to 185 cm corresponds
to (165 – 175)/10 and (185 – 175)/10, i.e., z = -1 to 1. The
probability required, as obtained from the statistical table, is
P (z < 1) – P (z <-1) ~ 68.3% (P stands for probability)
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only mean and sd
(x-m)/sd —> z score
find the probability between the two z score
z score can be apply no matter the distribution is normal or not
when the normal distribution is used, the z score can be used to l link the z score and percentile