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School of Psychology
Queen‟s University,
Belfast, BT7 1NN
1
Worked Examples of mathematics used in Psychology
Worked Example 1: Level 1 Introductory Psychology I (PSY1001) Laboratory
Demonstration
Worked Example 1 draws from CCEA GCSE Mathematics Module T1 and T2,
(Foundation Tier) – Statistics Component
For these exercises, describe in detail what the results presented in the tables and
graphs mean.
Answer:
This graph shows the mean extraversion and neuroticism scores for people grouped
by the number of friends they have. It shows that people with no friends have similar
levels extraversion and neuroticism, with a score of around 11 points. The trend for
extraversion shows that as the level of extraversion increases, so does the number
of close friends. This would be expected because extraversion is a personality trait
associated with outgoingness and sociability, so you would expect people who have
high levels of extraversion to have larger circles of friends. The opposite trend is
found with neuroticism, where lower levels of neuroticism are associated with having
more friends. This again is expected because neuroticism is a personality trait
associated with anxiety and worry. You would expect people with lower levels of
neuroticism to have larger circle of friends because people who are less prone to
anxiety and worry are probably able to make and maintain friendships.
NONE 1-2 3-5 5- 10+ 6
8
10
12
14
Mean EXTRA
Mean NEUROT
EXTRAVERSION AND NEUROTICISM
WITH NUMBER OF CLOSE FRIENDS
NUMBER OF CLOSE FRIENDS
Me
an
Score
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
2
EXTRAVERSION
NEUROTICISM
MEAN S.D. MEAN S.D.
MALE 12.20 5.30 7.27 4.38
FEMALE 11.53 4.34 10.36 5.00
Answer:
This table shows males are, on average, more extraverted compared to females with
males having a mean extraversion score of 12.2 and females having a score of
11.53. There is more variance in male extraversion scores, as males have a larger
standard deviation (5.3) compared to females (4.34). Females seem to be more
neurotic on average compared to males with a mean score of 10.36 compared to
7.27 for males. Neuroticism scores varied more for females, who have a larger
standard deviation compared to males.
8 12 16 20 24 28
0
10
20
30
FREQUENCY DISTRIBUTION
LIFE SATISFACTION INDEX - Z
LSI-Z SCORES
FR
EQ
UE
NC
Y
Answer:
This graph shows the distribution of standardised scores from the life satisfaction
questionnaire from a sample of approximately 100 people. The modal score on the
questionnaire was 24 with 29 people obtaining this score. The range of scores was
20 (with scores from 8 to 28). The shape of the distribution approximates a normal
distribution, with some evidence of negative skew seen in the tail to the left hand
side of the distribution.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
3
Most Enjoyed Least Enjoyed
1. Time spent with spouse (32%)
2. Doing things with children (22%)
3. Recreational activities/hobbies (18%)
4. Getting together with friends (13%)
1. Time spent alone (51%)
2. Taking care of the house (48%)
3. Recreational activities/hobbies (1%)
Answer:
This table shows a list of most and least popular past-times. The most popular
activity was spending time with spouse where 32% of respondents indicated they
most enjoyed this activity. Spending time alone was the least enjoyed activity with
51% of respondents indicating this. There appears to be some missing activities
rated as most enjoyed because the percentages of this group only adds up to 85%
meaning 15% o responses are unknown.
Background to this experiment: Participants were presented with a list of twenty
words and asked to remember as many words as possible. Each word was
presented one at a time on a computer screen for 5 seconds. Every participant saw
exactly the same list of words. After all 20 words were presented, the participants
were asked to write down as many words as they could remember from the list in
any order. The graph shows the percentage of participants who remembered each
word, depending on where the word appeared in the list. E.g. the word in word-
position 1 was presented first in the list, word position 10 is the middle word of the
list, and word position 20 was the final word presented.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
4
Answer:
This line graph shows the accuracy of recall for words presented as part of a list,
depending on each word‟s position in the list. It seems that the group were better at
remembering words presented at the start and the end of the list, compared to words
located in the middle of the list. Words at the end of the list were remembered better
than any other word positions, which would be expected because these words would
have been most recent in the participants‟ memory. Accuracy for words in the middle
of the list (approx. 40%) were approximately half that of the accuracy of recall for
words at the end of the list (approx. 80%).
Source: Questions based on level 1 laboratory handbook exercises, 2010/11.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
5
Worked Example 2: Level 1 Introductory Psychology 1 (PSY1001) Laboratory
Class Work
Worked Example 2 draws from CCEA GCSE Mathematics Modules:
T1 and T2, (Foundation Tier) – Mathematics & Algebra and Statistics
Components
T5 (Foundation Tier) -- Mathematics & Algebra Component
T3 (Higher Tier) – Statistics Component
T4 (Higher Tier) – Statistics Component
T6 (Higher Tier) – Mathematics & Algebra Component
GCSE Additional Mathematics:
Statistics, Topics 1 and 3
A-Level Modules:
S1, Topic 2
S2, Topics 5 and 6
Intelligence tests provide a standardised measure of ability commonly referred to as IQ.
IQ has a normal distribution centred on a mean score of 100 with a standard deviation of 15.
We are going to calculate the mean and standard deviation of IQ scores collected from a
sample of 20 people:
Participant Number IQ Score
1 110
2 124
3 99
4 108
5 117
6 136
7 96
8 107
9 101
10 112
11 120
12 95
13 114
14 126
15 100
16 119
17 121
18 108
19 101
20 113
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
6
Question:
We are going to use the below equation to calculate the mean IQ score of the group of
people:
Where:
is the symbol for mean
is a symbol that means „sum of/add up‟
is the symbol for individual IQ scores
N is the number of participants in the experiment
Fill in the values for and N in the equation below and calculate the mean IQ score:
Solution:
Now input the IQ scores to Microsoft excel and check your answer using the “=average”
function.
We will now calculate the standard deviation using the below equation:
Complete the table to help with the calculation:
Remember from above:
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
7
Participant Number IQ Score ( ) 1 110
2 124
3 99
4 108
5 117
6 136
7 96
8 107
9 101
10 112
11 120
12 95
13 114
14 126
15 100
16 119
17 121
18 108
19 101
20 113
___________
Solution:
Participant Number IQ Score ( ) 1 110
110-111.35 = -1.35 -1.35*-1.35 =
1.8225
2 124 12.65 160.0225
3 99 -12.35 152.5225
4 108 -3.35 11.2225
5 117 5.65 31.9225
6 136 24.65 607.6225
7 96 -15.35 235.6225
8 107 -4.35 18.9225
9 101 -10.35 107.1225
10 112 0.65 0.4225
11 120 8.65 74.8225
12 95 -16.35 267.3225
13 114 2.65 7.0225
14 126 14.65 214.6225
15 100 -11.35 128.8225
16 119 7.65 58.5225
17 121 9.65 93.1225
18 108 -3.35 11.2225
19 101 -10.35 107.1225
20 113 1.65 2.7225
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
8
2292.55
Now substitute the values into the standard deviation equation:
Solution:
10.985
Now check your answer using excel and/or PASW.
Question:
Compare the mean and standard deviation of this sample of 20 people with the mean and
standard deviation of the population of IQ scores ( = 100, s = 15) what could you conclude
about your sample?
Solution:
The sample of 20 people have a higher than average IQ at 111.35 This sample can‟t be
considered a representative sample of the population, where you would expect an average
IQ of around 100. The sample seems to be biased in favour of higher IQ scores. The
standard deviation is also lower than what would be expected in a representative sample at
10.985 as opposed to 15. This indicates there was less variance in IQ scores within the
group of 20 people than what you would expect if you were able to test the whole population.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
9
Question:
Does this histogram of the IQ data from the sample of 20 people approximate the normal
distribution that you would expect IQ data to take?
Solution:
Yes, the histogram shows that the IQ data closely fit the normal distribution.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
10
Worked Example 3: Level 2 Perception (PSY 2058) Laboratory Demonstration
and Coursework
Worked Example 3 draws from CCEA GCSE Mathematics Modules:
T5 (Foundation Tier) -- Mathematics & Algebra Component
T3 (Higher Tier) – Mathematics & Algebra Component and Statistics
Component
T4 (Higher Tier) – Statistics Component
T6 (Higher Tier) Mathematics & Algebra Component
GCSE Additional Mathematics:
Pure Maths, Topic 4
Statistics, Topic 6 and 7
A-Level Modules:
C2, Topic 4
S2, Topics 5 and 6
The speed/accuracy trade-off is used to describe „aimed movement‟ where someone is
required to move their hand from a starting point (S) to stop within a target area (T).
It describes how faster movements result in lower accuracy and higher accuracy can be
achieved by lowering speed of movement. Depending on the difficulty of a task, humans can
adjust their movement speed in order to achieve respectable accuracy.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
11
When the target area T is thin, the task is more difficult because movement requires higher
precision so as not to miss or overshoot the target. In this case movement must be slowed
down to hit the target accurately.
When the start-point and target are farther apart, the task difficulty is increased as the brain
needs to co-ordinate movement across a longer distance. In this case the longer distance
allows the arm to be accelerated to higher speeds, resulting in faster performance, however
accuracy may decrease as the extra deceleration of the arm may result in the target being
missed. This is why emergency buttons in aircraft cockpits are larger than other buttons. In
an emergency pilots will be working hard and quickly to avert disaster, so making the buttons
larger will make them easier to press with „fast and furious‟ hand movements.
The task difficulty index (Id) is a logarithmic function of the distance between the start-point
and target (amplitude of movement, A) and the width of the target (W) where:
Fitt‟s law states that there is a linear relationship between movement time (MT) and task
difficulty expressed as:
where m and c are constants.
An experiment was run where the amplitude and width of a target were manipulated across
16 trials. Movement Time was measured in milliseconds. Amplitude was measured at 4
levels (3, 6, 12 and 24 cm) and width was measured at 4 levels (1, 2, 4 and 8 cm). The
results of each trial are in the table below:
Trial Number Amplitude (cm) Width (cm) Movement Time (ms)
1 3 1 360
2 3 2 212
3 3 4 182
4 3 8 166
5 6 1 217
6 6 2 353
7 6 4 179
8 6 8 172
9 12 1 234
10 12 2 383
11 12 4 325
12 12 8 229
13 24 1 481
14 24 2 404
15 24 4 338
16 24 8 298
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
12
Question: Input this table into Excel and calculate, in a new column, the Index of Task
Difficulty (Id) for each trial. Use the log function where „Number‟ is the expression and
„Base‟ is 2.
Solution:
Assuming the above table is inputted in Excel where the top-left cell “Trial Number” is A1;
the excel expression for item difficulty for trial 1 will be: =log(2*B2/C2,2). This formula can be
filled down to calculate Id for the other rows, which are given in the following table.
Trial Number Movement Time (ms) Task Difficulty
1 360 2.584963
2 212 1.584963
3 182 0.584963
4 166 -0.41504
5 217 3.584963
6 353 2.584963
7 179 1.584963
8 172 0.584963
9 234 4.584963
10 383 3.584963
11 325 2.584963
12 229 1.584963
13 481 5.584963
14 404 4.584963
15 338 3.584963
16 298 2.584963
Question:
Does the data support the hypothesis that Fitt‟s Law provides an adequate description of the
role assumed by movement amplitude and precision in determining movement time?
Solution:
Since Fitt‟s law is a linear equation, correlation can be used to calculate the extent of the
linear relationship, using the square of the correlation coefficient as an indication of common
variance.
∵
[PASW or Excel can be used to calculate the correlation coefficient.]
There is a strong correlation between movement time and difficulty index (r=.7698) where
59.2% of the variance in movement time is accounted for by the difficulty index variance.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
13
Question:
What is the correspondence of the slope and intercept values obtained from a regression
analysis of MT and Id to the parameters m and c in the equation:
Solution:
Using PASW, you can perform a regression analysis with a predictor of Id and a criterion
variable of MT. The relevant analysis output is given the table:
Coefficientsa
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig. B Std. Error Beta
1 (Constant) 164.986 30.735 5.368 .000
Id 45.775 10.143 .770 4.513 .000
a. Dependent Variable: MT
The unstandardised beta coefficient for Id is the slope of the regression line which
corresponds to the gradient (m) parameter of the Fitt‟s law equation. The unstandardised
beta coefficient of the constant is the intercept of the regression line, which corresponds to
the intercept (c) parameter of the Fitt‟s law equation.
Substitution of the constants gives the solution:
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
14
Question:
Draw a scatter plot showing Movement Time (y-axis) and Index of Difficulty (x-axis); include
the line of best fit as described in the regression.
Solution:
Source: Questions based on level 2 laboratory demonstration and coursework from 2008.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
15
Worked Example 4: Level 2 Methods and Conceptual Issues (PSY 2056)
Research Methods and Statistics
Worked Example 4 draws from CCEA GCSE Mathematics Modules:
T1 and T2, (Foundation Tier) – Mathematics & Algebra and Statistics
Components
T5 (Foundation Tier) -- Mathematics & Algebra Component
T3 (Higher Tier) – Statistics Component
T4 (Higher Tier) – Statistics Component
T6 (Higher Tier) – Mathematics & Algebra Component
GCSE Additional Mathematics:
Statistics, Topic 3
A-Level Modules:
S1, Topic 1 and 2
S2, Topics 3 and 4
Question:
As a clinical psychologist, you want to investigate the effectiveness of a new type of therapy
for depression. Before starting the new therapy, you ask clients to complete the Beck‟s
Depression Inventory. This is self-report questionnaire that provides a measure of
depression on a scale from 0-63 where 0 is minimal depression and 63 is severe
depression. After two months of therapy, you ask clients to complete the depression
inventory again to see if levels of depression have improved during the course of the
therapy. Twenty clients completed the therapy course, their inventory scores are below.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
16
Client Identification Number
Beck‟s Depression Inventory Score BEFORE Therapy
Beck‟s Depression Inventory Score AFTER two months
1 44 23
2 13 17
3 58 55
4 7 7
5 32 19
6 33 45
7 28 30
8 37 28
9 41 30
10 23 18
11 52 42
12 11 8
13 32 40
14 59 54
15 8 13
16 60 34
17 29 25
18 46 42
19 15 11
20 33 28
Assuming the therapy is effective; state your one-tailed hypothesis for the depression
scores.
Solution:
It is hypothesised that clients will score significantly lower in Depression scores after two
months of therapy, compared to their scores just before starting the therapy.
Question:
Looking at the table, do you think the therapy is effective?
Solution:
Generally, there seems to be lower scores after the therapy. The mean depression score for
the whole group before treatment was 33.05, whereas after treatment the mean for the
group reduced to 28.45. This implies that depression scores reduced over the course of the
treatment. However, there are a few clients whose scores increased after the therapy
(clients 2, 6, 7, 13, 15). The difference between scores also varied across clients, for
example, client 16‟s scores reduced by 26, client 7 only reduced by 2. In conclusion, the
treatment seems to work as hypothesised for some people but it is still unclear if the effect of
the treatment is large enough to be considered effective.
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
17
You can perform a t-test to compare the mean depression scores of the group before and
after therapy. By using a t-test you can also decide if the difference in scores is statistically
significant. The formula for a t-test is:
Where is the average difference between the depression scores before and after
treatment and is the standard error of the mean difference scores which is given by:
Where is the standard deviation of the difference scores and N is the number of clients
tested.
Question:
Calculate the t-test using excel to complete column D in following table:
Column A Column B Column C Column D
Client Identification
Number
Beck‟s Depression Inventory Score
BEFORE Therapy
Beck‟s Depression Inventory Score
AFTER two months
Difference in depression score:
Column B – Column C
1 44 23
2 13 17
3 58 55
4 7 7
5 32 19
6 33 45
7 28 30
8 37 28
9 41 30
10 23 18
11 52 42
12 11 8
13 32 40
14 59 54
15 8 13
16 60 34
17 29 25
18 46 42
19 15 11
20 33 28
Mean of Difference Scores :
Standard Deviation of Difference Scores :
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
18
Solution:
Column A Column B Column C Column D
Client Identification
Number
Beck‟s Depression Inventory Score
BEFORE Therapy
Beck‟s Depression Inventory Score
AFTER two months
Difference in depression score:
Column B – Column C
1 44 23 44-23 = 21
2 13 17 -4
3 58 55 3
4 7 7 0
5 32 19 13
6 33 45 -12
7 28 30 -2
8 37 28 9
9 41 30 11
10 23 18 5
11 52 42 10
12 11 8 3
13 32 40 -8
14 59 54 5
15 8 13 -5
16 60 34 26
17 29 25 4
18 46 42 4
19 15 11 4
20 33 28 5
Mean of Difference Scores : 4.6
Standard Deviation of Difference Scores : 9.058
Substitute the values and calculate SEM from the above equation:
Solution:
School of Psychology
Queen‟s University,
Belfast, BT7 1NN
19
Now, calculate t by substituting the values to the equation:
Solution:
Values of t greater than 1.96 are statistically significant at p<.05 level. I.E. there is a 5%
chance or less that the difference between the mean of the two groups is due to chance. Is
the difference in mean depression scores before and after therapy significant at the p<.05
level?
Solution:
Yes, t=2.27 which is greater than the threshold value of 1.96