Summation Notation, Percentiles and Measures of Central Tendency

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Summation Notation, Percentiles and Measures of
Central Tendency

• Overheads 3

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Statistical Notation for Variables
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Organizing Your Data
Grp 1 Grp 2
Obs 1 9.00 7.00
Obs 2 5.00 8.00
Obs 3 4.00 10.00
Obs 4 3.00 1.00
Obs 5 2.00 14.00
Grp 1 Grp 2
Obs 1 X1 1 X1 2
Obs 2 X2 1 X2 2
Obs 3 X3 1 X3 2
Obs 4 X4 1 X4 2
Obs 5 X5 1 X5 2
X4 13.00 X5 12.00 X3 210.00 X5 214.00
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Sigma Notation

• Often, it is necessary for us to add together
  sets of scores, so we need a convenient way to
  tell someone Add up the scores for a group of
  people.
• In statistics, the greek symbol sigma is used to
  denote add together.

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Summation Notation if there is only one group.
means... Sum the raw scores for i1 to N
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Example
Grp 1
10.00
9.00
11.00
12.00
7.00
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Summation Notation for more than one group.
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Example
Grp 1 Grp 2
3.00 4.00
4.00 2.00
1.00 5.00
2.00 6.00
1.00 3.00
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Order of Operations
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In-class Statistical Notation Problem
Set (located in Course Materials)
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Problem 1
99.00
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Problem 2
(5.006.004.00)
(7.001.002.00)


25.00
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Problem 3
5.00 6.00 4.00 3.00 2.00 20.00
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Problem 4
5.00 7.00 9.00 4.00 25.00
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Problem 5
5.00 7.00 9.00 4.00 25.00
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Problem 6
(52 25) (6236) (4216) 611.00
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Problem 7
(5 6 4 3 2)2 202 400
(7 1 2 5 8)2 232 529
(9 10 7 2 3)2 312 961
(4 5 6 7 3)2 252 625
2,515
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Problem Set
is different from
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Shapes/Types of Distributions
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Shapes/Types of Distributions
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Shapes/Types of Distributions
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Shapes/Types of Distributions
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How can we divide up the frequency distribution.

• Percentiles
• A frequency distribution divided into 100 equal
  parts.
• A percentile tells us what percent (proportion of
  the distribution) falls at or below the score
  interval of interest.
• Quartiles
• A frequency distribution divided into four equal
  parts.
• Q1 P25 Q2 P50 Q3 P75 Q4 P99
• Deciles
• A frequency distribution divided into 10 equal
  parts.
• D1, D2, D3, , D10 P99
• All of these measures are on ordinal scales.

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Percentiles and the Normal Distribution
These are not equivalent halves!
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Note See Handout Location of Percentiles on a
Normal Curve in Course Materials
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Percentiles and the Normal Distribution
This line must be moved to the left to form two
equivalent halves!
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Note See Handout Location of Percentiles on a
Normal Curve in Course Materials
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Quartiles and the Normal Distribution
25
25
25
25
Q1
Q2
Q3
P75
P25
P50
X
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Deciles and the Normal Distribution
D5 100 D4 99.5 D3 99 D2 98 D1 96.5
.50
.50
1.00
1.50
99.5
96.5
98
99
100
99
98
97
96
D5
D4
D3
D2
D1
X
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Getting a percentile rank for a particular raw
score.
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Getting a raw score for a specific percentile.
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Measures of Central Tendency

• Measures of central tendency help to give
  information about the most likely score in a
  distribution.
• We have three ways to describe central tendency
• Mean
• Median
• Mode
• The type of measure of central tendency you
  should use depends on what kind of data you have.

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The Mode

• The Mode is the score within a set of scores that
  appears most frequently.
• The Mode is appropriate for Nominal scale data.
• If all scores are the same then there is no Mode.
• If two adjacent scores both have the same, and
  the highest frequency, then the Mode is the
  average between the two scores.
• If two non-adjacent scores have the same and
  highest frequency then the group of scores is
  Bimodal.

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The Mode
X f X f
8.00 4 10.00-11.00 8
7.00 6 8.00-9.00 12
6.00 10 6.00-7.00 (midpoint 6.5) 21
5.00 8 4.00-5.00 17
4.00 5 2.00-3.00 9
3.00 2 0.00-1.00 2
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The mode
MODE
MODE
MODE
MODE
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Median

• The Median is the 50th percentile in a group of
  scores.
• The Median divides the rank scores so that half
  of the scores fall above the median and half fall
  below.
• The Median is calculated exactly as the 50th
  percentile.

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The median
MEDIAN
MEDIAN
MEDIAN
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Finding the median for an ungrouped frequency
distribution.

• If there is an odd number of scores then the
  median is the middle score.
• If there is an even number of scores then the
  median is the halfway point between the middle
  most two values.

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Finding the median for an ungrouped frequency
distribution.

• N35 (odd number of scores)
• N35/2 17.5
• Since we do not have half scores, we use the
  18th scores to represent the median.

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The median.
X f
8.00 4
7.00 6
6.00 10
5.00 8
4.00 5
3.00 2

• There are a total of 35 scores, so we are looking
  for the interval with the 18th score.
• The cumulative frequency reaches 18 in the
  interval of 6.00, therefore, the median is 6.00.

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The median

• There are a total of 16 scores, so we are looking
  for the that has the two middle scores (the 8th
  and 9th scores).
• The 8th score is in the interval 5.00 and the 9th
  score is in the interval 6.00. So, the median is
  5.50.

X f
8.00 1
7.00 2
6.00 5
5.00 5
4.00 2
3.00 1
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The Mean

• Mean
• Mean of combined groups when nj is equal for all
  groups
• Mean of combined groups when nj is not equal for
  all groups

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Practicing Calculations Measures of Central
Tendency

• See Handout in Course Materials

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In-class exercise Measures of central
tendency (located in Course Materials)
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Properties of the mean.

• 1) The sum of all deviation scores around the
  mean will be exactly zero.

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Properties of the mean.
See handout Properties of the mean Located
in Course Materials
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Properties of the mean.

• The sum of all deviation scores around the mean
  will be exactly zero.
• The sum of squared deviations will always be less
  than the sum of the squared deviations around any
  other point.
• Least sum of squares.

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The mean
MEAN
MEAN
MEAN
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Location of Mean, Median, and Mode in a
Distribution

• If a distribution is symmetrical, and unimodal,
  the mean, median and mode will have the same
  value.
• If a distribution is unimodal and skewed, these
  measures will be arranged in the order of mean,
  median, and mode, starting from the longest tail.
• In negatively skewed distributions the mean will
  be less than the median.
• In positively skewed distribution the mean will
  be greater than the median.
• The difference between the mean and the median in
  a distribution is an indication of skewness.

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The mean, median, and mode.
Mode
Mean Median
Median
Mean
Mode
Mode
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Central Tendency for Normal Distribution
Mean 15.00
Median 15.00
Mode 15.00
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Central Tendency for Bimodal Distribution
Mean 15.00
Median 15.00
Mode 14.00 and 16.00
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Central Tendency for Positively Skewed
Distribution
Mean 13.10
Median 12.00
Mode 12.00
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Central Tendency for Negatively Skewed
Distribution
Mean 16.8966
Median 18.0000
Mode 18.00
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SPSS-Calculating measures of central tendency
Change var names to group names
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SPSS-Calculating measures of central tendency
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SPSS-Calculating measures of central tendency
We can run a single group (as shown) or all four
groups at a time
To get measures
of central tendency, click Statistics
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SPSS-Calculating measures of central tendency
To find the raw score that corresponds to
the 65th percentile, (1) check box, (2) type in
percentile, (3) click add
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SPSS-Calculating measures of central tendency

• See Handout for Output for Central Tendency in
  Course Documents

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SPSS-Calculating measures of central tendency
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SPSS-Calculating measures of central tendency
If there is more than 1 mode, SPSS reports the
lowest one and tells you other modes exist
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SPSS-Calculating measures of central tendency
An alternate way of obtaining the
measures of central tendency is with
Descriptives
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SPSS-Calculating measures of central tendency
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SPSS-Calculating measures of central tendency
The Descriptives table puts
the group variables in rows and statistics in
columns
The means of each group
The minimums and maximums are the lowest
and highest scores in each group
The means
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SPSS-Calculating measures of central tendency
A third option for obtaining the measures of
central tendency is with Explore
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SPSS-Calculating measures of central tendency
Transfer all four group variables to the
dependent list and click ok
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SPSS-Calculating measures of central tendency
Explore provides the mean and the median, not the
mode