Download MEASURES OF DISPERSION

MEASURES OF DISPERSION
HANDOUT 5.4
Range
Definition: The difference between maximum and minimum.
Calculation:
1 Arrange data into ascending array
2 Identify the minimum maximum values
3 Calculate the range (MAXV – MINV)
Interquartile Range
Definition: the difference between the 75th percentile (75% of the
data) and the 25th percentile (25% of the data) and includes the
median, or 50th percentile.
Represents the central portion of the normal distribution
Calculate:
1 Arrange data in increasing order
2 Find position of first and third quartiles
Q1 = (n+1)/4
Q3 = 3(n+1)/4 = 3xQ1
3 Identify the values (Whole numbers match the observations,
fractions lie between observations)
4 Interquartile range is Q3-Q1
Example: Data set is 13, 7, 9, 15, 11, 5, 8, 4
STEP 1: Arrange the array in ascending order
4, 5, 7, 8, 9, 11, 13, 15
STEP 2: Determine Q1 position
= (n+1)/4
= (8+1)/4 = 2.25
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STEP 3: Count observations from the beginning of the array. The
calculated amount of 2.25 means the second item in the array plus .25
times the difference between 2nd and 3rd observations.
= 5 + .25 x (7-5)
= 5.5
STEP 4: Determine Q3 position
= 3(n+1)/4
= 3(9)/4
= 6.75
STEP 5: Repeat Step 3 procedures
The calculated amount of 6.75 is the sixth item plus .75 times
the difference between the 6th and 7th observation.
= 11 + ¾(13-11)
= 11 + ¾(2)
= 12.5
The Interquartile Range (IQR) = Q3 - Q1
= 12.5 - 5.5
=7
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Variance
Definition: Variance (s² or ²) is a measure of dispersion around the
mean of a distribution.
Calculate:
1 Arrange the data into ascending order
2 Create a frequency distribution table with
column headings for X‾, X, (X‾-X), (X
‾-X)²
X = value
X
‾ = mean
(X
‾-X) = difference from the mean
(X
‾-X)2 = difference squared
3 Sum the (X
‾-X)² column
The formula is (s²) = (X
‾-X)²/n-1
n = total observations
Standard Deviation
Definition: The standard deviation (s or ), is another measure of
dispersion around the mean of a distribution. It indicates how the
data falls within the curve of the frequency distribution
Calculate: The standard deviation is the square root of the variance.
The formula is s = (X
‾-X)²/n-1
Approximately 68% of the values will occur within (+/-) 1 standard
deviation (1s) of the mean.
Approximately 95% of the data will occur within (+/-) 2 standard
deviations (2s) of the mean.
Fully 99.7 % of the data will occur within (+/-) 3 standard deviations
(3s) of the mean.
Values which are more than 2 standard deviations from the mean are
only 5% of the total data - a figure that is considered by most
researchers to be the cut- off point for "statistical significance."
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Choosing the appropriate measures of central
tendency and dispersion for frequency distributions
Normal distribution: The mean is the preferred measure of central
tendency. The standard deviation is the preferred measure of
dispersion.
Skewed distribution: The median is the preferred measure of
central tendency. The interquartile range is the preferred measure of
dispersion.
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