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BIOSTATISTICS

TOPIC 5.4MEASURES OF DISPERSION

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BIOSTATISTICS

• TERMINAL OBJECTIVE:

• 5.4 Calculate Measures of Dispersion.

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Enabling Objective

• E.O. 5.4.1 State the purpose of determining measures of dispersion.

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Measures of Dispersion

• Purpose– To describe how much spread there is

in a distribution.

– Used with a particular measure of central tendency

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Enabling Objectives

FROM A SET OF STATISTICAL DATA, COMPUTE THE:

5.4.2 Range.5.4.3 Interquartile range.5.4.4 Variance.5.4.5 Standard deviation

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Range

• Definition– The difference between maximum

and minimum.

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Range

Calculation

• Arrange data into ascending array• Identify the minimum maximum values• Calculate the range

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Interquartile Range

• Defined: 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

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100th percentile

25th 75th percentile percentile

median

Q1 Q3

Interquartile range = Q3 - Q1

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Interquartile Range

• Calculate Interquartile range from individual data– Arrange data in increasing order– Find position of first and third

quartiles • Q1 = (n+1)/4 • Q3 = 3(n+1)/4 = 3xQ1

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Interquartile Range

• Calculate Interquartile range from individual data– Arrange data in increasing order– Find position of first and third

quartiles • Q1 = (n+1)/4 • Q3 = 3(n+1)/4 = 3xQ1

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Interquartile Range

– Identify the values– Whole numbers match the

observations.– Fractions lie between observations – Interquartile range is Q3-Q1

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Interquartile Range

• Example: Observations- 13, 7, 9, 15, 11, 5, 8, 4 STEP 1: Arrange the array4, 5, 7, 8, 9, 11, 13, 15 STEP 2: Determine Q1 position= (n+1)/4 = (8+1)/4 = 2.25

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Interquartile Range

STEP 3: Count observations from beginning of array

2.25 is the second plus ¼ difference between 2nd and 3rd observations

= 5 + ¼(7-5) = 5.5

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Interquartile Range

STEP 4: Determine Q3 positionQ3 = 3(n+1)/4= 3(9)/4= 6.75 STEP 5: Repeat Step 3 procedure to

locate value in array

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Interquartile Range

6.75 is the sixth plus ¾ difference between the 6th and 7th observation= 11 + ¾(13-11)

= 11 + ¾(2) = 12.5

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Interquartile Range

IQR = Q3-Q1 = 12.5 - 5.5 = 7

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Variance

• Variance (s²) is a measure of dispersion around the mean of a distribution.

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Variance

• Calculate Variance from Ungrouped Data – Arrange the data into ascending order

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Variance• 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

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Variance

• Sum the (X-X)² column• Formula: (s²) = (X-X)²/n-1n = total observations

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Standard Deviation

• The standard deviation, s, is the square root of the variance.

– s = (X-X)²/n-1

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Standard Deviation

• Indicates how the data falls within the curve of the frequency distribution– Approximately 68% of the values will

occur within (+/-) 1 standard deviation (1s) of the mean X.• X ± 1s = 68%

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Standard Deviation

– Approximately 95% of the data will occur within (+/-) 2 standard deviations (2s) of the mean X • X ± 2s = 95%

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Standard Deviation

– 99.7 % of the data will occur within (+/-) 3 standard deviations (3s) of the mean• X ± 3s = 99.7%

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Standard Deviation

– Values which are (+/-) 2s 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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Enabling Objective

E.O. 5.4.6 State the appropriate measure of dispersion for frequency distributions

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Choosing Measures Of Dispersion

• Normal distribution– The standard deviation is preferred

• Skewed distribution – The interquartile range is preferred

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