Basic Statistics

1
Basic Statistics
Measures of Central Tendency
2
Characteristics of Distributions

• Location or Center
• Can be indexed by using a measure of central
  tendency
• Variability or Spread
• Can be indexed by using a measure of variability

3
Consider the following distribution of scores
How do the red and blue distributions differ?
How do the red and green distributions differ?
4
Consider the following distributions
How do the green and blue distributions differ?
5
Consider the following two distributions
How do the green and red distributions differ?
6
Characteristics of Distributions

• Location or Central Tendency
• Variability
• Symmetry
• Kurtosis

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Measures of Central Tendency
Summarizing Data
The Mean The Median The Mode
Give you one score or measure that represents, or
is typical of, an entire group of scores
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Most scores tend to center toward
a point in the distribution.
frequency
score
Central Tendency
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Measures of Central Tendency
Are statistics that describe typical, average, or
representative scores.
The most common measures of central tendency
(mean,median, and mode) are quite different in
conception and calculation. These three
statistics reflect different notions of the
center of a distribution.
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The Mode
The score that occurs most frequently
In case of ungrouped frequency distribution
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Unimodal Distribution -One Mode-
Bimodal Distribution Two Modes-
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Mode and Measurement Scales
Can you find a mode for each data?
Nominal Scale
Ordinal Scale
Interval Scale
Ratio Scale
1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2
1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4
3
4
112
56
68 56 39 56 44 56 45 56 75 81 67 59
112 132 112 113 112 150 125 114
Nationality 1American 2Asian 3Mexican
Football Poll 1first 2second 3third 4fourth
IQ score
Weight
13
The Mode

• It is not affected by extremely large or small
  values and is therefore a valuable measure of
  central tendency when such values occur.
• It can be found for ratio-level, interval-level,
  ordinal-level and nominal-level data

14
The Median
The Median is the 50th percentile of a
distribution - The point where half of
the observations fall below and half of the
observations fall above In any distribution there
will always be an equal number of cases above and
below the Median.
Oh my !! Where is the median?
Location
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For an odd number of untied scores (11, 13,
18, 19, 20)
11 12 13 14 15 16 17 18 19
20
The Median is the middle score when scores are
arranged in rank order
Median Location (N1)/2 3rd
Median Score 18
16
For an even number of untied scores (11, 15,
19, 20)
11 12 13 14 15 16 17 18 19
20
The Median is halfway between the two central
values when scores are arranged in rank order
Median Location (N1)/2 2.5th Score
Median (1519)/2 17
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• The Median of group of scores is that point on
  the number line such that sum of the distances of
  all scores to that point is smaller than the sum
  of the distances to any other point.
• There is a unique median for each data set.
• It is not affected by extremely large or small
  values and is therefore a valuable measure of
  central tendency when such values occur.

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The Median

• Can be computed for
• Ordinal-level data
• Interval-level data
• Ratio-level data

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Median and Levels of Measurement
1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2
3 2
1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4
112 132 112 113 112 150 125 114
68 56 39 56 44 56 45 56 75 81 67 59
No
Yes
Yes
Yes
Nationality
Football Poll
IQ score
Weight
Can you find a median for each type of data?
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The Mean
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The Population Mean

• For ungrouped data, the population mean is
  the sum of all the population values divided by
  the total number of population values. To
  compute the population mean, use the following
  formula.

Sigma
Individual value
Population mean
Population size
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The Sample Mean

• For ungrouped data, the sample mean is the sum
  of all the sample values divided by the number of
  sample values. To compute the sample mean, use
  the following formula.

Sigma (Summation)
Sample Mean
Individual value
Sample size
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Characteristics of The Mean
Center of Gravity of a Distribution
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Center of Gravity of a Distribution
1
2
3
4
5
6
7
8
Mean
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How much error do you expect for each
case?
Deviation Scores
-6
25
27
-4
31
31
-2
0
29
33
2
The Mean
6
35
37
4
Data set
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On average, I feel fine
Its too hot!
Its too cold!
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The Mean of group of scores is the point on the
number line such that sum of the squared
differences between the scores and the mean is
smaller than the sum of the squared difference to
any other point. If you summed the differences
without squaring them, the result would be zero.
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• Mean and Measurement Scales
• Every set of interval-level and ratio-level data
  has a mean.
• Can you find the Mean for the following data
  sets?

Nominal data
Ordinal data
Interval data
Ratio data
1 2 3
1 2 3
1 2 3
1 2 3
2
2
YES
NO
YES
NO
Nationality 1American 2Asian 3Mexican
IQ Test
Football Poll 1first 2second 3third
Weight
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• All the values are included in computing the mean.

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• A set of data has a unique mean and the mean is
  affected by unusually large or small data values
  outliers.

3
5
7
9
1
1
9
3
5
6
5
4
5
5.5
5
The Mean
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• Every set of interval-level and ratio-level data
  has a mean.
• All the values are included in computing the
  mean.
• A set of data has a unique mean.
• The mean is affected by unusually large or small
  data values.
• The arithmetic mean is the only measure of
  central tendency where the sum of the deviations
  of each value from the mean is zero.

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The Relationships between Measures of Central
Tendency and Shape of a Distribution
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Normal Distribution
Symmetric
Unimodal
Mean Median Mode
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Positively Skewed Distribution
Mode
Median
Mean
Mode lt Median lt Mean
The median falls closer to the mean than to the
mode
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Negatively Skewed Distribution
Mode
Median
Mean
Mode gt Median gt Mean
The median falls closer to the mean than to the
mode
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Bimodal Distribution
Mode1
Mode2
Mean Median
Mode1 lt Mean Median lt Mode2
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SUMMARY
There are three common measures of central
tendency. The mean is the most widely used and
the most precise for inferential purposes and is
the foundation for statistical concepts that will
be introduced in subsequent class. The mean is
the ratio of the sum of the observations to the
number of observations. The value of the men is
influenced by the value of every score in a
distribution. Consequently, in skewed
distributions it is drawn toward the elongated
tail more than is the median or mode. The median
is the 50th percentile of a distribution. It is
the point in a distribution from which the sum of
the absolute differences of all scores are at a
minimum. In perfectly symmetrical distributions
the median and mean have same value. When the
mean and median differ greatly, the median is
usually the most meaningful measure of central
tendency for descriptive purposes. The mode,
unlike the mean and median, has descriptive
meaning even with nominal scales of measurement.
The mode is the most frequently occurring
observation. When the median or mean is
applicable, the mode is the least useful measure
of central tendency. In symmetrical unimodal
distribution the mode, median, and mean have the
same value.