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Transcript of 1 Descriptive statistics: Measures of dispersion Mary Christopoulou Practical Psychology 1 Lecture...
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Descriptive statistics:
Measures of dispersion
Mary Christopoulou
Practical Psychology 1 Lecture 3
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Key terms
Standard Deviation= τυπική / “σταθερή”
απόκλιση από τον μέσο όρο
Variance/ variability = Διακύμανση
Range = Εύρος
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Descriptive statistics:
statistical measures that summarize and
communicate the basic characteristics
of a distribution
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2 Types of Descriptive Statistics
Measures of central tendency:measures that communicate the degree to
which scores are centred in a distribution
Measures of dispersion:measures that communicate the degree to
which scores are spread out in a
distribution
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Measures of
central tendency Mean
Median
Mode
Measures of
dispersion Range Interquartile range Variance Standard Deviation
2 Types of Descriptive Statistics
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Measures of dispersion
They show: how far from the center the data tend to
range/ spread. the extent to which scores in a distribution
differ from each other How large are the differences between
individual scores? How much variability is there in the data?
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An example
N of chocolate cookies consumed by 10 girls and 10 boys:
Girls = 5, 3, 9, 12, 10, 12, 4, 9, 7, 2 (mean = 7.3) Boys = 6, 7, 7, 8, 8, 8, 7, 7, 8, 7 (mean = 7.3)
Which of the two means is a more accurate reflection of underlying data? 2 issues:
consistency of data: smaller amount of variability indicates a greater consistency of the data (and vice versa)
Accuracy/reliability of measure of central tendency
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Example cntd…
Girls = 5, 3, 9, 12, 10, 12, 4, 9, 7, 2 (mean = 7.3)
Boys = 6, 7, 7, 8, 8, 8,7, 7, 8, 7 (mean = 7.3)
Girls’ mean is based on scores with greater variability,
and Boys’ mean is based on scores with smaller
variability.
Mean for Boys is a more accurate reflection of
underlying data, as it is based on a sample that is
more consistent (from one score to the next).
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Range
Range is the difference (distance)
between the highest and lowest value
in the data
Can be calculated for all levels of
measurement, apart from the nominal
level.
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RANGE example
RANGE = 9 - 1 = 8
1 2 3 4 52 7 9
3, 2, 4, 9, 5, 7, 1, 2
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How to calculate the range
1. Put scores from lowest through highest2. Find the highest value and the lowest
value of the data set 3. Subtract the lowest score from the
highest score
Example: 22, 25, 30, 42, 88, 102Range is 102 – 22 = 80
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Range
Used as a very quick (“rough”) method
Quite inefficient – only the smallest and
largest values are used
Excessively vulnerable to outliers
(extreme scores)
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Inter Quartile Range
The IQR (Inter Quartile Range) is not affected by extreme scores
Divide the data into 4 equal parts (Quartiles)
delete the extreme quarters of data and measure range of middle 50%
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28 2 4 1 5 2 7
How to find the IQR:1. Put scores in order 2. Delete the extreme quartiles
1 2 2 4 5 7 28
IQR= 5 - 2 = 3
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3
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Variance and Standard Deviation
Two measures of variability that tell us how much scores are spread around a mean.
Note: a mean of 50 could indicate that most
scores are between 48-52, or could indicate
that most scores are between 40 and 80!!
48 - - - 52
40 - - - - - - - - - - - - - - - - - - - - - - - - - - - - 80
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Variance (or Variability)
It is the degree to which scores are spread
around the mean
It involves the average square deviation of
each value from the mean of the values
It is the average error between the mean and the
observations made.
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Example Which set of scores are more spread out?
Set A: 40, 40, 50, 60, 60 mean=50
Set B: 40, 49, 50, 51, 60 mean=50
40 50 60
40 60
40 50 60
49 51
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If we subtract each score from its group
mean we can see that:
4 of the scores in set A are 10 units away from
the mean,
whereas only 2 scores in set B are 10 units
from the mean.
Thus A has the greater variance!!
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Variance
The most efficient of the measures of dispersion
Only valid for interval & ratio scales
Vulnerable to outliers
The larger the value of the Variance, the more
each score is “distant” from the mean.
The smaller the Variance, the closer each score
is, to the mean.
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The Standard Deviation (SD)
Recall that calculation of the Variance involves squaring the deviation scores.
This means the Variance value is much larger than the actual deviation of scores from the mean.
Therefore, the variance value is not reported often.
Instead, report the Standard Deviation.
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Standard Deviation
Is the average amount by which scores in a
distribution differ from the mean Shows the
average distance of the data from the mean
It is a measure of how well the mean
represents the data.
Is a measure of the degree of dispersion of
the data from the mean.
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Interpreting SD…
large SD = scores are far from the mean (so, the mean is not an accurate representation of the data).
small SD = scores are closer to the mean (scores are more
clustered around the mean).
E.g. 3 data sets, each has an average of 7.
1. 0, 0, 14, 14 SD = 7
2. 0, 6, 8, 14 SD = 5
3. 6, 6, 8, 8 SD = 1
The 3rd set has a much smaller standard deviation than
the other 2 because its values are all close to 7.
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Interpreting SD using our example…
Recall that mean = 5, SD = 2.00 in our example
“Boundary at minus one SD” = 5 - 2 = 3 “Boundary at plus one SD” = 5 + 2 = 7 These boundaries indicate how much scores
are spread around the mean. Thus, the majority of scores in our sample are
between 3 and 7 (this is true)
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Important!
In scientific reports (including your lab write-ups) it is important to report both a measure of central tendency and a measure of dispersion
Mean & SD for normally distributed data Median & IQR for skewed data
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Reporting Results APA format
Q. what is the mean and standard deviation? A. (M = 5.00, SD = 2.00).
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Small Revision
Descriptive Statistics
Measures of Central Tendency
Measures of Dispersion
Range
Interquartile Range
Variance
Standard Deviation