2.5 measure of position

College of Engineering Engineering Statistics
Department of Dam & Water Resources Lecturer: Goran Adil & Chenar
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Measure of Position 2.5
Chapter 2: Part 5: Measure of Position
Objectives
 Determine and interpret z-scores
 Determine and interpret percentile
 Determine and interpret quartile
 Check a set of data for outliers
5.1: Z-Score (or standard score)
The number of standard deviations that a given value x is above or below the mean.
The following formula allows a raw score, x, from a data set to be converted to its equivalent
standard value, z, in a new data set with a mean of zero and a standard deviation of one.
A z-score can be positive or negative:
 positive z-score – raw score greater than the mean
 negative z-score – raw score less than the mean
Whenever a value is less than the mean, its corresponding z score is negative
Ordinary values: z score between –2 and 2 sd
Unusual Values: z score < -2S.D or z score > 2 S.D
x x
z
s


x
z




Sample Population

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Example 2.21 Annual rainfall.
If the annual rainfalls in a certain city are 1, 22, 26, 33, and 123. cm over a 5-year period.
Determine the z-score for each raw score. Discuss the value ?. is there any UN-USUAL Data

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5.2: Position
There are several ways of measuring the relative position of a specific member of a set.
5.2.1: Quartiles
 Quartiles split a set of ordered data into four parts.
 Q1 is the First Quartile
 25% of the observations are smaller than Q1 and 75% of the observations are
larger
 Q2 is the Second Quartile
 50% of the observations are smaller than Q2 and 50% of the observations are
larger. Same as the Median. It is also the 50th percentile.
 Q3 is the Third Quartile
 75% of the observations are smaller than Q3and 25% of the observations are
larger
Q1, Q2, Q3 .Divides ranked scores into four equal parts
The lower quartile is the median of the lower half of the data and the upper quartile is the
median of the upper half.
The median divides the data in the data in half. The upper and lower quartiles divide each half
into two parts.

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Example 2.22
Find Q1, Q2, and Q3 for the following data set.
15, 13, 6, 5, 12, 50, 22, 18
Sort in ascending order.
5, 6, 12, 13, 15, 18, 22, 50
 1
6 12
Q , 9
2
median Low MD

  
 2
13 15
Q , 14
2
median Low High

  
 3
18 22
Q , 20
2
median MD High

  
5.2.2: Percentiles
Just as there are quartiles separating data into four parts, there are 99 percentiles denoted P1, P2, .
. . P99, which partition the data into 100 groups.
 A percentile is the value of a variable below which a certain percent of observations fall.
So the 20th percentile is the value (or score) below which 20 percent of the observations may be
found.
A person with percentile rank of 20, means that he /she scored the same as or better than 20
percent of the group.
0 5
100


.number of values less than x
Percentile of valuex
total number of values

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Example 2.23
The following series is the minimum monthly flow (m3 S-l
) in each of the 10 years Of a certain
river.
36, 4, 21, 21, 23, 11, 10, 10, 12, 17
- What percentage of the data is less than 11?
- What percentage of the data is less than 23?

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5.2.3: Converting from the kth Percentile to the Corresponding Data Value
n total number of values in the data set
k percentile being used
L locator that gives the position of a value
Pk kth percentile
HINT
 Step 3: If L is not a whole number, round up to the next whole number. If L
is a whole number, use the value halfway between L and L+1.
100
k
L n

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Example 2.24
A teacher gives a 20-point test to 10 students. Find the value corresponding to the 25th
and 60th
percentile.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
a) For 25th
percentile
The value 5 corresponds to the 25th
percentile.
(a student who had 5, did better than 25th
percent of all student)
b) For 60th
percentile
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
7 the vale = 12
Hence, A score of 11 correspond to the 60th
Percentile
100
k
L n
25
10 2 5 3
100
  .L roundupto
In part A the L value was not a
whole number (2.5)
Hence, L was rounded up to the
next large number (2.5 rounded up
to 3).
And the corresponding value to
the 25th
percentile is the 3rd
from
the lowest
100
k
L n
60
10 6
100
 L
In Part B the L value is a whole
number (6)
FROM THE HINT
If L is a whole number, use
the value halfway between L
and L+1.
10 12
11
2

average
6th
value
7th
value

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5.3: Exploratory Data Analysis
Exploratory Data Analysis is the process of using statistical tools (such as graphs, measures of
centre, and measures of variation) to investigate data sets in order to understand their important
characteristics
5.3.1: Outlier
An outlier is a value that is located very far away from almost all the other values.
Important Principles
 An outlier can have a dramatic effect on the mean
 An outlier have a dramatic effect on the standard deviation
 An outlier can have a dramatic effect on the scale of the histogram so that the true nature
of the distribution is totally obscured
5.3.2: Box-lots(Box-and-Whisker Diagram)
A box-and-whisker plot shows the spread of a data set. It displays 5 key points: the
minimum and maximum values, the median, and the first and third quartiles.
A box-plot is a graph of the five-number summary.
 A central box spans the quartiles;
 A line in the box marks the median;
 Lines extend from the box out to the smallest and largest observations.
 Useful for side-by-side comparison of several distributions.

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Example 2.25
on Making a Box-and-Whisker Plot and Finding the Interquartile Range
Make a box-and-whisker plot of the data. Find the interquartile range.
{6, 8, 7, 5, 10, 6, 9, 8, 4}
Step 1 Order the data from least to greatest.
4, 5, 6, 6, 7, 8, 8, 9, 10
Step 2 Find the minimum, maximum, median, and quartiles.
Step 3 Draw a box-and-whisker plot.
Draw a number line, and plot a point above each of the five values. Then draw a box from the
first quartile to the third quartile with a line segment through the median. Draw whiskers from
the box to the minimum and maximum.
IRQ = 8.5 – 5.5 = 3
The interquartile range is 3, the length of the box in the diagram.
Step 1 Order the data from least to greatest.
11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23
Step 2 Find the minimum, maximum, median, and quartiles.

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Method of detecting an outlier
 To determine whether a data value can be considered as an outlier:
 Step 1: Compute Q1 and Q3.
 Step 2: Find the IQR = Q3 – Q1.
 Step 3: Compute (1.5)(IQR).
 Step 4: Compute Q1 – (1.5)(IQR) and
Q3 + (1.5)(IQR).
 Step 5: Compare the data value (say X) with Q1 – (1.5)(IQR) and Q3 + (1.5)(IQR).
 If X < Q1 – (1.5)(IQR) or
if X > Q3 + (1.5)(IQR), then X is considered an outlier.
Example 2.26
 Given the data set 5, 6, 12, 13, 15, 18, 22, 50, can the value of 50 be considered as an
outlier?

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Tutorial on 2.5
Examples 2.17
The following series is the minimum monthly flow (m3 S-l
) in each of the 20 years 1957 to 1976
at Bywell on the River Tyne: 21, 36, 4, 16, 21, 21, 23, 11, 46, 10, 25, 12, 9, 16, 10, 6, 11, 12, 17,
and 3
- Find Q1, Q2, and Q3 for the following data set.
- Determine the z-score for each raw score. Discuss the value ?. is there any UN-USUAL Data
- What percentage of the data are less than 21?
- What percentage of the data are less than 36?
- Find the value corresponding to the 25th
and 60th
. 91th
percentile.
- Make a box-and-whisker plot of the data. Find the interquartile range.
Example 2.18
A data set has a mean of 10 and a standard deviation of 2. Find a value that is:
(i) 3 standard deviations above the mean
(i) 2 standard deviations below the mean
Example 2.19
Find Q1, Q2, and Q3 for the data set.
121, 129, 116, 106, 114, 122, 109, 125.
Example 2.20
Given a data set with a mean of 10 and a standard deviation of 2, determine the z-score for each
of the following raw scores,. [8,10,16].
Exercise 2.21
Compute the quartiles from the following data.

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Examples 2.22
Make a box-and-whisker plot of the data. Find the interquartile range. {13, 14, 18, 13, 12,
17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23}
Exercise 2.23
Compute the quartiles from the following data.
Marks No. of students
1-10 3
11-20 16
21-30 26
31-40 31
41-50 16
51-60 8
Exercise 2.24
Stream flow velocity. A practical example of the mean is the determination of the mean velocity
of a stream based on measurements of travel times over a given reach of the stream using a
floating device. For instance, if 10 velocities are calculated as follow:
Velocity,
m/s
0.20 0.20 0.21 0.42 0.24 0.16 0.55 0.70 43 0.34
- What percentage of the velocities are less than 0.24 m/sec?
- What percentage of the data are less than 0.34 m/sec?
- Find the velocity value corresponding to the 25th
, 51th
and 70th
. 91th
percentile.
Exercise 2.25
Make a box-and-whisker plot of the data. Find the interquartile range.
{6, 8, 7, 5, 10, 6, 9, 8, 4}

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Exercise 2.26
Make a box-and-whisker plot of the data. Find the interquartile range. {13, 14, 18, 13, 12,
17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23}
Exercise 2.28
The data values on the table below depict the maximum monthly discharges of a certain River
for nine consecutive months. Create a box-and-whisker plot to display the data.
April May June July August September October November December
110, 98, 91, 102, 89, 95, 108, 118, 152
Exercise 2.29
Concrete cube test. From 28-day concrete cube tests made in England in 1990, the following
results of maximum load at failure in kilonewtons and compressive strength in newtons per
square millimeter were obtained:
Maximum load: 950, 972, 981, 895, 908, 995, 646, 987, 940, 937, 846, 947, 827,
961, 935, 956.
Compressive strength: 42.25, 43.25, 43.50, 39.25, 40.25, 44.25, 28.75, 44.25, 41.75,
41.75, 38.00, 42.50, 36.75, 42.75, 42.00, 33.50.
- What percentage of the Maximum load are less than 895 and 950 KN?
- What percentage of the compressive strength are 42.00 less than ?
- Find the maximum load and compressive strength values corresponding to the 25th
, 51th
and
70th
. 91th
percentile.
- Make a two separate box-and-whisker plot for the two set of data.. Find the
interquartile range.
Exercise 2.30
An experiments measuring the percent shrinkage on drying of 50 clay specimens produced the
following data :
18.5 22 24 19 16 21.5 17 15
20 10 9.5 17 20.5 19 22.5

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A) Compute the sample and population for both variance and standard deviation
B) Draw box-plot and indicate outlier
Exercise 2.31
The data in Table below (Adamson, 1989) are the annual maximum flood peak flows to the
Hardap Dam in Namibia, covering the period from October1962 to September 1987. The range
of these data is from 30 to 6100.
Annual maximum flood-peak inflows to Hardap Dam (Namibia): catchment area 12600 km2
Year 1962-3 1963-4 1964-5 1965-6 1966-7 1967-8 1968-9 1969-0 1970-1
Inflow (m3 S-l) 1864 44 46 364 911 83 477 457 782
Year 1971-2 1972-3 1973-4 1974-5 1975-6 1976-7 1977-8 1978-9 1979-0
Inflow (m3 S-l) 6100 197 3259 554 1506 1508 236 635 230
Year 1980-1 1981-2 1982-3 1983-4 1984-5 1985-6 1986-7
Inflow (m3 S-l) 125 131 30 765 408 347 412
- What percentage of the discharges are less than 911 m3
/sec?
- What percentage of the data is less than 1864 m3
/sec?
- Find the discharge value corresponding to the 25th
and 60th
. 91th
percentile.

2.5 measure of position

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