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42 3 Summarizing Data
In this example, the albumin level is a quantitative continuous value, and since the
limits of the classes are given by an integer, we subtract 0.5 units from the lower limit
of the median class (40) to calculate the actual boundary for the median calculation.
The main difference between the mean and the median
Indeed, the calculation of the mean involves the actual values of the data, whereas
the median relies solely on their ranks, disregarding the actual values. Consequently,
extremely large or small values exert a lesser influence on the median compared to
the mean. The mean’s formula incorporates every value in the dataset, making it
sensitive to outliers, whereas the median’s formula hinges only on the middle value
or values, thereby offering more resilience against the impact of outliers. This aspect
is especially crucial when analyzing datasets containing outliers.
In skewed distributions where the mean is affected by very large or very small
values (such as income), the median is a more desirable index.
Since the median divides data into two equal parts, it is also called the second
quartile (
Q
2
) or the fifth decile (D
5
) or the fiftieth percentile (C
50
). Therefore, by
generalizing the above formula, percentiles, deciles, and quartiles can be calculated.
Quartiles
Quartiles are points that divide the dataset into four equal parts. There are 3 quartiles,
which are called the first quartile (
Q
1
), the second quartile (Q
2
) and the third quartile
(Q
3
) and determine the points of 25, 50 and 75% of the data distribution, respectively.
Their computation formula is like the median formula.
Q
1
= L
i
+
1 ×
N +1
4
− F
i−1
f
i
× w
i
Q
2
= L
i
+
2 ×
N +1
4
− F
i−1
f
i
× w
i
Q
3
= L
i
+
3 ×
N +1
4
− F
i−1
f
i
× w
i
,
where:
L
i
is the lower class boundary of the class containing the related quartile.
N is the total number of data.
F
i−1
is the cumulative frequency of precede class of the related quartile class that
should be excluded.
f
i
is the frequency of the related quartile class.
w
i
is the class width of the related quartile class.
Example 6
Find the first and third quarters of the female albumin level data in
Table
3.2.
Solution: To find the third quarter ( j = 3) and according to the number of samples,
the location of the third quarter is in the rank
3×(N +1)
4
=
3×(185+1)
4
= 139.5.The
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3.2 Measures of Central Tendency 43
cumulative frequency of the class (43–45) is immediately greater than that. Therefore,
L
i
= 42.5, f
i
= 48, w
i
= 3, F
i−1
= 112. Then by substitution the above values in
the formula, we have:
Q
3
= L
i
+
3 ×
N +1
4
− F
i−1
f
i
× w
i
= 42.5 +
3 ×
185+1
4
− 112
48
× 3 = 44.22.
The exact value of the third quarter is equal to 44.22, which means that 75% of the
observed values are less than 44.22 and only 25% of them will be more than 44.22.
Similarly, to calculate the first quarter, L
i
= 36.5, f
i
= 52, w
i
= 3, F
i−1
= 17 and
by substitution the above values in the formula, we have:
Q
3
= L
i
+
1 ×
N +1
4
− F
i−1
f
i
× w
i
= 36.5 +
1 ×
185+1
4
− 17
52
× 3 = 38.20.
The exact value of the first quarter is equal to 38.2, which means that 75% of the
observed values are less than 38.2 and only 25% of them will be more than 38.20.
Deciles
Deciles are points that divide the dataset into 10 equal parts. There are 9 deciles, which
are called the first decile (
D
1
), the second decile (D
2
) … and the ninth decile (D
9
)
and determine the points of 10, 20, …. and 90% of the data distribution, respectively.
Their computation formula is similar to the quartile’s formula.
D
1
= L
i
+
(
1×
N+1
10
−F
i−1
)
f
i
× w
i
D
2
= L
i
+
(
2×
N+1
10
−F
i−1
)
f
i
× w
i
.
.
.
.
.
.
D
9
= L
i
+
(
9×
N+1
10
−F
i−1
)
f
i
× w
i
,
where:
L
i
is the lower class boundary of the class containing the related decile.
N is the total number of data.
F
i−1
is the cumulative frequency of precede class of the related decile class that
should be excluded.
f
i
is the frequency of the related decile class.
w
i
is the class width of the related decile class.
https://t.me/med1917
44 3 Summarizing Data
Example 7 Find the ninth decile of the female albumin level data in Table 3.2.
Solution: To find the third quarter ( j = 9) and according to the number of samples,
the location of the third quarter is in the rank
9×(N +1)
10
=
9×(185+1)
10
= 167.4.The
cumulative frequency of the class (46–48) is immediately greater than that. Therefore,
L
i
= 45.5, f
i
= 23, w
i
= 3, F
i−1
= 160. Then by substitution the above values in
the formula, we have:
D
9
= L
i
+
9 ×
N +1
10
− F
i−1
f
i
× w
i
= 45.5 +
9 ×
185+1
10
− 160
23
× 3 = 46.46.
The exact value of the first quarter is equal to 46.46, which means that 75% of the
observed values are less than 46.46 and only 25% of them will be more than 46.46.
Centiles
Centiles are points that divide the dataset into 100 equal parts. There are 99 deciles,
which are called the first centile (
C
1
), the second centile (C
2
) … and the ninety
ninth centile (
C
99
) and determine the points of 1, 2 …. and 99% of the data
distribution, respectively. Their computation formula is similar to the quartiles and
deciles formula.
C
1
= L
i
+
(
1×
N +1
100
−F
i−1
)
f
i
× w
i
C
2
= L
i
+
(
2×
N +1
100
−F
i−1
)
f
i
× w
i
.
.
.
.
.
.
C
99
= L
i
+
(
99×
N+1
100
−F
i−1
)
f
i
× w
i
,
where:
L
i
is the lower class boundary of the class containing the related centile.
N is the total number of data.
F
i−1
is the cumulative frequency of precede class of the related centile class that
should be excluded.
f
i
is the frequency of the related centile class.
w
i
is the class width of the related centile class.
Example 8
Table 3.3 determines the level of urea acid in 186 women based on the
contraceptive pill. Get the 20th percentile.
Solution: To find the third quarter ( j = 20) and according to the number of samples,
the location of the third quarter is in the rank
20×(N +1)
100
=
20×(185+1)
100
= 37.4.The
cumulative frequency of the class (30–39) is immediately greater than that. Therefore,
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3.2 Measures of Central Tendency 45
Table 3.3 Frequency distribution of urea acid in 186 women in terms of contraceptive pill
consumption
Urea acid Consumed pill Not consumed To t a l Cumulative freq
20–29 3 4 7 7
30–39 20 22 42 49
40–49 42 43 85 134
50–59 19 13 32 166
60–69 5 9 14 180
70–79 1 1 2 182
80–89 3 0 3 185
90–99 1 0 1 186
Total 94 92 186 –
L
i
= 29.5, f
i
= 42, w
i
= 10, F
i−1
= 7. Then by substitution the above values in the
formula, we have:
C
20
= L
i
+
20 ×
N +1
100
− F
i−1
f
i
× w
i
= 29.5 +
20 ×
186+1
100
− 7
42
× 3 = 36.74.
The exact value of the twentieth centile is equal to 36.74, which means that 20%
of the observed values are less than 36.74 and only 80% of them will be more than
36.74.
The corresponding points of median, quartiles, deciles, and percentiles can be
specified by Ogive plot (cumulative frequency curve) was determined. This can be
done by the following steps:
Specify the desired quantile on the Y -axis.
From that point, draw a line parallel to the X-axis to intersect the Ogive plot.
From the intersection point, draw a vertical line to the X-axis.
The point of collision of this line (M ) with the X-axis is the value of the desired
index (Fig.
3.2).
Midpoint Range (MPR)
The midpoint range is another measure of central tendency that has less usage than the
median and mean. This index is denoted by MPR and is the average of the minimum
and maximum values, i.e.:
MPR =
X
Max
+ X
Min
2
,
where the X
Max
and X
Min
are the maximum and minimum values, respectively. It is
named sometimes mid-range (MR).
https://t.me/med1917
46 3 Summarizing Data
I
F
E
A
B
M
N
D
G
O
N/2
K
C
Midpoint Class
Ogive graph
1
Q
Q3
Median
Decile
9th
50%
90%
75%
25%
Cumulative
freq
Fig. 3.2 Geometrical interpretation of the median. AB = Class width of median class, AD = BC
= Cumulative frequency of precede class of median class, N = Total frequency, CE = Absolute
frequency of median class, BE = AD + CE = Lower boundary of median class, OA = L =
Difference between median value and lower boundary in median class, =
FC =
N
2
− BC
Example 9 The number of pain relief pills used by 40 patients in one day is as
follows. Get the midpoint range of pain relief for these 40 patients (Table
3.4).
Solution: X
Min
= 0, X
Max
= 12 and the midpoint range for these 40 patients is:
MPR =
X
Max
+ X
Min
2
=
1211 + 0
2
= 6.
Mode
Mode is the value of an observation which has the highest frequency (peak) and is
denoted by Mod. This index is not unique and may have two or more modes in a
dataset (distribution). If a distribution has a single mode, it is called a “unimodal”
distribution, if it has two modes, it is called a “bimodal” distribution, and if it has
several views, it is called a “multimodal” distribution (Fig.
3.3).
Special attention should be paid to the distribution with more than one mode,
because it may be possible to better understand and interpret its data by separating
the subgroups (Fig.
3.4).
Determining and calculating the mode in raw data does not require a special
formula. Because after sorting the data, the value that has most frequency is selected
as the mode. But the value of the mode in the frequency tables and categorized data is
Table 3.4 Number of pain relief pills taken daily in 40 patients
6, 11, 2, 3, 11, 8, 9, 11, 4, 3, 7, 12, 4, 3, 0, 1, 8, 2, 8, 4, 9, 4, 6, 9, 5, 4, 4, 5, 3, 2, 4, 8, 1, 6, 8, 9, 3,
6, 5, 1
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3.2 Measures of Central Tendency 47
Fig. 3.3 Example of multimodal distribution
Fig. 3.4 Separated subgroups of Fig. 3.3
obtained using proportionality and interpolation, according to the following formula:
Mod = L
i
+
d
b
d
b
+ d
a
× w
i
,
where
L
i
is the lower class boundary of the modal class.
d
b
is the frequency difference between the modal class and the previous one.
d
a
is frequency difference between the modal class and the next one.
w
i
is the class width.
Tip
The modal class is the class that has the largest ratio of “absolute frequency”
to “class width”.
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48 3 Summarizing Data
Fig. 3.5 Geometrical
interpretation of the mode
a
d
b
d
Mod
Table 3.5 Age distribution
of population in a
hypothetical village
Age groups Freq
Less than 1 Y 36
1–5 155
6–10 185
11–14 110
15–24 215
25–64 340
65+ 42
Sum 1083
In tables with equal class width, the modal class is the class that has the highest
absolute frequency. But in tables with unequal class width, the modal class is the class
whose has the largest ratio of “absolute frequency” to “class width”. This geometric
interpretation of the mode is shown in the figure below (Fig.
3.5).
Example 10
Table 3.5 shows t he age distribution of the population in a village. Find
out the mode of this population.
Solution: To calculate the mode, you must first find the modal class. When the class
width is equal, the modal class is the same class with the highest frequency. However,
if the class width is not equal, the modal class is a class with the highest ratio of
absolute frequency to the class width. Therefore, we add another column to the table
above to show the ratio of absolute frequency to class width.
According to the fourth column of Table 3.6, the third class 6–10 years is the
modal class (the gray class). Since the studied variable of age is a discrete variable,
the lower boundary of this class is equal to the lower class limit. Therefore, we
substitute the values of
w
i
= 5, L
i
= 6, d
a
= 185−155 = 30, d
b
= 185−110 = 75
in the mode formula.
Mod = L
i
+
d
b
d
b
+ d
a
× w
i
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3.2 Measures of Central Tendency 49
= 6 +
30
30 + 75
× 5 = 7.43.
This means that in this village, the highest frequency is related to the age of 7.43
(7–8) years old.
In most distributions, the following empirical relationship is settled between the
mean, median, and mode. By having the median and mean, the value of the mode
can be obtained.
μ − Mo = 3
(
μ − Med
)
or Mo = 3Med − 2μ.
To calculate the central tendencies (mean, median, mode, and centiles) with SPSS,
choose the path of analyze\descriptive statistics\frequencies to open a new window.
Now select the desired variables from the left box and move to the right box. Then
click the statistics option to open another window. At the top right of this window,
mark the mean, median, and mode options.
In the percentile values box, the first option (quartiles) to calculate the quartiles, the
second option (cut points for …. Equal groups) is for data cut points for equal groups.
So, in this example, it specifies 5 groups with equal numbers and the percentile(s)
option is assigned to the specific percentiles that the user writes the desired percentile
and moves to the below box by pressing the add button.
Table 3.6 Necessary calculations for Table
3.5 to find the mode
Age groups Freq Class width Ratio of frequency to the class width
Less than 1Y 36 1 36/1 = 36
1–5 155 5 155/5 = 31
6–10 185 5 185/5 = 37
11–14 110 4 110/4 = 27.5
15–24 215 10 215/10 = 21.5
25–64 340 40 340/140 = 8.5
65+ 42 35 42/35 = 1.2
Sum 1083 – –
https://t.me/med1917
50 3 Summarizing Data
3.3 Measures of Dispersion
Sometimes two or more datasets which have very close measures of central tendency
may differ from each other. The similarity of these measures, without paying attention
to their differences raises interpretive problems. For example, consider the birth
weight data of two samples of 5 newborn babies as follows:
A: 2.70, 2.90, 3.50, 3.90, 4.10.
B: 3.00, 3.20, 3.45, 3.65, 3.80.
Despite their means and medians are very close together (Med
A
= 3.50, Med
B
=
3.45,μ = 3.42
), these two datasets are different and not similar. Also look at the
weight of the following two samples of 10.
C: 65, 66, 67, 68, 71, 73, 74, 77, 77, 77.
D: 100, 93, 85, 77, 77, 67, 62, 58, 54, 52.
Although the two datasets appear to be different, the measures of central
tendency are exactly equal to
μ = 71.5, MPR = 71, Mod = 77 and Med = 72.
Therefore, the measures of central tendency alone are not able to provide a clear
and comprehensive glance of the data and other indicators should be used to reveal
these differences. The following figure further illustrates this necessity (Fig.
3.6).
The measures that describe the spread and variation of the dataset are called
“measures of dispersion”, which we described below.
Fig. 3.6 Different frequency distributions with identical means
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3.3 Measures of Dispersion 51
Range
The range (R) represents the most straightforward measure of dispersion in a dataset,
determined by subtracting the smallest observation value from the largest.
R = X
Max
− X
Min
.
The value of R is always non-negative, because the smaller number must be
subtracted from the larger number. It is very simple to calculate because it only
involves and is affected by two values. It is also very sensitive to extreme values. So,
the index is very weak because its value varies only by changing the two values of
X
Max
and X
Min
observations. If the rest of the data change, its value will not change.
Interquartile range
The interquartile range (IQR) serves as a measure of spread and is invariably non-
negative, calculated as the difference between the first and third quartiles.
IQR = Q
3
− Q
1
,
where Q
1
and Q
3
are the first and third quartiles, respectively. IQR is a useful
indicator to display the middle 50% of the data distribution.
Example 11 Find the interquartile range for data on female albumin levels in
Table
3.2.
Solution: In example 6, the first and third quartiles of that table were 38.20 and
44.22. So the interquartile range is:
IQR = Q
3
− Q
1
= 44.22 − 38.20 = 6.02
This means that the distance between Q
1
and Q
3
(middle 50% of the data
distribution) is 6.02 units (Fig.
3.7).
Employing the interquartile range (IQR) to detect outlier values in a dataset is a
widely used and effective method. To achieve this, subtract 1.5 times the IQR from
the first quartile and add 1.5 times the IQR to the third quartile to determine the upper
1
Q
2
Q
IQR=Q
3
-Q
1
Upper outlier boundary
=Q
3
+1.5IQR
Lower outlier boundary
=Q
1
-1.5IQR
Outliers
3
Q
Fig. 3.7 Location of quartiles, median, and IQR
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