numerical descriptive measures cont d · 2/16/2017 · 5 measures of positions: percentiles a...

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Numerical Descriptive Measures Cont’d

Measures of Central Tendency (Location)

The Arithmetic Mean (Average)

The Median

The Mode

Measures of Dispersion or Variability

The Range

The variance

The Standard Deviation

The Coefficient of Variation

Measures of Relative Standing (Measures of Position)

Z Score

Quartiles and Percentiles

STA 231: Biostatistics

Dr. Ahmed Jaradat, AGU2/12/2017

Measures of Relative Standing (Measures of Position)

In some cases, the analysis of certain individual items in the data set is of

more interest rather than the entire set.

It is necessary at times, to be able to measure how an item fits into the

data, how it compares to other items of the data, or even how it

compares to another item in another data set.

Measures of position are several common ways of creating such

comparisons

Z Score (or Standard Score) The number of standard deviations that a given value x is above or below the

mean, the Z-score is given by:

Population σ

μxZ ,

Sample

xZ ,

s

X

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Dr. Ahmed Jaradat, AGU

Example: Comparing Z-Scores

Two students, who take different medical classes, had exams on the same day.

Khalid’s score was 83 while Hassan’s score was 78. Which student did relatively

better, given the class data shown below?

Hassan’s z-score is higher as He was positioned relatively higher within

his class than Khalid was within his class.

Khalid

Class

Hassan

Class

Class mean 78 70

Class standard deviation 4 5

25.14

873

8Z sKhalid' 61

5

7078.Z sHassan'

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Measures of Positions: Percentiles

50% 50%

Median

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Dr. Ahmed Jaradat, AGU 4

25% 25% 25% 25%

Q1 Q2=Median Q3

10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

D1 D2 D3 D4 D5=Q2 D6 D7 D8 D9

1% 1% … … 1% 1%

P1 P2 P50=Q2 P98 P99

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Measures Of Positions: Percentiles

A percentile is the score at which a specified percentage of scores in a

distribution fall below.

The percentile rank of a score indicates the percentage of scores in

the distribution that fall at or below that score.

Percentile (Pr)

The rth percentile of a set of measurements is the value for which

At most r% of the measurements are less than that value.

At most (100-r)% of all the measurements are greater than that value.

Example

Suppose 600 is the 78% percentile (P78) of a GMAT score. Then

600200 800

78% of all the scores lie here 22% lie here



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Commonly used percentiles

First (lower) Decile = 10th percentile

First (lower) Quartile, Q1 = 25th percentile

Second (middle) Quartile,Q2 = 50th percentile = Median

Third Quartile, Q3 = 75th percentile

Ninth (upper) Decile = 90th percentile

Location of Percentiles:

Find the location of any percentile using the formula

If the result is a whole number then it is the ranked position to use.

If the result is a fractional half (e.g. 7.5, 11.5, 26.5, etc.) then average the two corresponding data values.

If the result is not a whole number or a fractional half then interpolate between the data points.



percentile thr the of location theisr

Lwhere

100

r1)(n

rL

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Find the first quartile and the median of the following set of measurements (n=15):

7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th

2, 4, 4, 5, 7, 8, 8, 10, 12, 17, 18, 21, 27, 29, 30

P25,

Q1= 4th observation= 5

P50,

Q2=Median= 8th observation= 10



Example

4.0100

251)(15L25

0.8100

50)115(L50

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Find the median and third quartile of the following set of measurements (n=16)

7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8, 40

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th

2, 4, 4, 5, 7, 8, 8, 10, 12, 17, 18, 21, 27, 29, 30 40

P50,

P50 =Q2=Median= (8th observation+ 9th observation )/2

= (10+12)/2=11

P75,

P75 =Q3= 12th value+0.75(13th value –12th value)

= 21 +0.75(27-21)=25.5



Example

5.8100

501)(16L50

7512100

7511675 .)( L

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Calculate the 30th, 67th, 90th percentile of the following set of measurements (n=15):

7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th

2, 4, 4, 5, 7, 8, 8, 10, 12, 17, 18, 21, 27, 29, 30

P30,

P30 = 4th observation+ 0.8(5th observation - 4th observation) = 6.6

P67,

P67 =10th observation+ 0.72(11th observation - 10th observation)= 17.72

P90 =29.4STA 231: Biostatistics


Example

4.8100

301)(15L30

7210100

6711567 .)( L

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IQR= P75 – P25

= Q3 – Q1

This helps to get a range that is not influenced by the extreme high and low scores.

Where the range is the spread across 100% of the scores, the IQR is the spread across the middle 50%.

Large value indicates a large spread of the observations.

The IQR is a measure of variability that is not influenced by

outliers or extreme values

Measures like Q1, Q3, and IQR that are not influenced by outliers

are called resistant measures

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Interquartile Range, IQR

Box plots

A box plot is a graphical display that provides the main descriptive

measures of the measurement set.

To construct a box plot, we need only five statistics:

1. The minimum value,

2. Q1(The first quartile),

3. The median,

4. Q3 (The third quartile), and

5. The maximum value.

Outliers: An observation x is called an outlier if:

x < Q1 - 1.5 (IQR) or x > Q3 + 1.5 (IQR)

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The Box Plot

Median

(Q2) maximumminimumQ1 Q3

Example: Draw a for the data:

11 12 13 16 16 17 18 21 22

25% 25% 25% 25%

11 12.5 16 19.5 22

Interquartile range

= 19.5 – 12.5 = 7

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Distribution Shape and The Box plot

Positively-SkewedNegatively-Skewed Symmetrical

Q1 Q2 Q3 Q1 Q2 Q3Q1 Q2 Q3

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S =41,

L =100,

Q1=66.5 ,

Q2=76,

Q3=89,

IQR=89-66.5

=22.5

Interpreting the box plot results

The scores range from 41 to 100.

About half the scores are smaller than 76, and about half are larger than 76.

About half the scores lie between 66.5 and 89.

About a quarter lies below 66.5 and a quarter above 89.

An outlier is any grade x < 66.5-1.5 (22.5)=32.75 or x > 89.0+1.5 (22.5)=122.75

Example: Consider the Biostatistics grades

numerical descriptive measures cont d · 2/16/2017 · 5 measures of positions: percentiles a...

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