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Descriptive Measures of Data
“Battalions of figures are like battalions of men, not always as strong as
is supposed”
- M.Sage

Learning Objectives
Given the learning materials and activities of this chapter, you will be able to
? 1. find the mean, median and the mode to describe the center of a set of data;
? 2. calculate the range, the mean deviation, the variance and the standard
deviation to describe the variability of a set of data;
? 3. determine ranks as measures of position for a given set of data;
? 4. describe the shape of a distribution using measures of skewness and kurtosis;
? 5. interpret values of specific descriptive measures of data and evaluate their
relative merits and limitations.

? The Summation Symbol (Σ)
?The summation notation uses the Greek capital word Σ
(sigma) to denote the sum of values.

i. Arithmetic Mean (Mean)
? The arithmetic mean, or simply, the mean of a set of data is the sum
of the values divided by the number of values.
Where X is the values of a data set.

Example:
? The ratings given by five judges to a painting exhibit using a 10-point rating scale,
with 10 as excellent and 1 as poor, were as follows: 8, 5, 9, 8 and 7. Treating the
data as a population, what is the mean rating of the five judges?
Xn = 8, 5, 9, 8, 7

a. Population and Sample Mean
? The sample mean is the average of a subset (sample) of
observations taken from a larger population. The population mean
is the average of all observations in a specific population.

Sample Mean e.g.
? The scores of a sample of 11 undergraduate students in a 100-item final
examination in Statistics were recorded as follows: 70, 83, 74, 75, 81, 75, 92, 75,
90, 74 and 95. Find the mean for the given data.

ii. The Median
? The median of a set of data, arranged in an increasing or decreasing order
magnitude, is the value that splits the data into two halves such that half of the
values are greater than or equal to the median and the other half of the values
are less than or equal to the median.
if n is odd: if n is even:
(n+2)/2

Example (Sample Mean e.g.)
? The recorded sample: 70, 83, 74, 75, 81, 75, 92, 75, 90, 74, 95
? To find the median, we first arrange the scores in an array, that is, in
an increasing order of magnitude, as follows: 70, 74, 74, 75, 75, 75,
81, 83, 90, 92,95. Since number of observations (n = 11) is odd, the
median is determined as follow:
Median =X (n + 1)/2 =X (11+1)/2 =X6 =75
? Since X6 is 75 in the given data set.

Example (Even) :
The ages of 10 Grade I teachers at the City Central School are as follows: 31,36, 42,
42, 55, 57, 57, 59, 60 and 62. Find the median age.
? Solution: Since n = 10 and the given ages are arranged in an increasing order,
the median is obtained as follows:

iii. Mode
? The mode is the value that occurs most frequently in a set of data.
For ungrouped data, it is easy to determine the mode and does not
require any calculation.
For example, the given data set from the 10 students from Biliran
International University who took the numeracy test; the data shows
their scores out of 100 items test; 70, 81, 95, 70, 65, 90, 70, 75, 91, 70
For the given example, the mode is 70 since it appeared four times.
? Another example, given data set; 10, 12, 11, 20, 10, 14, 11, 27
There are two modes which is 10 and 11 who appeared twice on the
data set and this is called a bimodal.

I. The Appropriateness of an Average
The measures of central tendency the mean, median and the
mode are measure of the average. To decide which statistical
average is most appropriate for a given se of data, the nature of
the data and the use to which the average is to be applied an
important considerations. Also, we shall consider the relative merits
and limitations each of these statistical measures.

ii.
Aside from the fact that the arithmetic mean is a simple and familiar measure, it has
following desirable properties:
1. It can be calculated from any set of numerical data, so it always exists.
2. A set of data has one and only one mean, so it is always unique.
3. It takes into consideration every item in the set of data.
4. It lends itself to further statistical treatment. For instance, the means of a number
of separate groups of data can readily be combined into an overall mean, called
the grand mean, without going back to the original data
5. It is relatively reliable and stable in sampling. The means of many samples drawn
from the same population usually do not fluctuate, or vary, as widely as other
statistics used to estimate the population mean.

iii.
? An outlier is either an extremely large or extremely s value in a set of
data which can affect the mean to such an extent that it may give
distorted impression of the data.
? The median always exists, and unique and is also easy to find once
the data have been arranged in the order of magnitude.
? One advantages of the median is that it is not affected by extreme
values and not sensitive to change the values of the items that
make up the set of data.

iv.
? The two main advantages of the mode are: (1) it requires no
calculation, and (2) it can be used for qualitative as well as
quantitative data. However, a set of data may have either no
mode or more than one mode. It is an unstable measure and its
value may not be representative of the center of a given set of
data. Thus, the mode is not widely used.

Measures of Variability: Range
?The range is the difference between the largest and
smallest values in a set of data.
Formula: Range
Range = (Largest Value) – (Smallest Value)

Example of Range:
? In a 100-item test, the set of data is gained as follows; 92, 90, 86, 54,
48, 31
Solution:
Range = 92 – 31 = 61,
thus, the range of the given data set is 61.

The Mean Deviation
? Also called the average deviation, the mean deviation of a set of
value x1, x2, …, xn is defined as the arithmetic mean of the absolute
values of their deviations from their arithmetic mean.
Formula: Mean Deviation (Ungrouped)
MD =

Example:
? The scores of a random sample of 5
Grade 11 pupils on a 10-item spelling
quiz were 3, 5, 2, 8 and 7. Find the
mean deviation of the sample.
Solution:
Mean x =
=
=5
MD =
=
=
=2

Measures of Position
Ranks, Fractiles and Quantiles

i. Measures of Position
?Aside from the measures of central tendency and
variability, measures of position are used to describe the
position of a score or any observation relative to the entire
set of data. A measure of position describes the ordinal
position of any score in a population of ordinal of interval
ratio data.

ii. Ranks
?The rank of a score or any observation in a
population is a measure of position that reflects
the absolute number of the score in that
population whose values fall above that score. It is
an index of the absolute number of scores above
a given score in a population. (Trooper, 1998)

a. Formula 1: Assigning Ranks to Scores
? The rank assigned to any given score is the number of scores in the
population greater than that score plus one(1).
Rank = A + 1
Where A is the number of scores greater than the score being ranked.

Formula 2: Assigning Ranks to Tied Scores
? When two or more individuals in a population have the same score,
their score are said to be tied. A group of individuals whose scores
are equal to one another is called a knot.
Rank = A+
Where A is the number of scores above the knot and B is the number
of scores in the knot.

Example:
Student Score
Ariel 35
Ben 39
Dick 32
Ella 42
Fred 45
Jay 40
Mia 35
Olga 35
Rod 40
Ted 38
This table shows the scores of ten graduate in a 50-item Proficiency
Test. Rank These students on the basis of their scores.
To facilitate ranking, the students
may be assigned based on their
scores from the highest to lowest.
If we used the first formula which
is the Rank = A + 1,
Rank of Fred = A + 1
= 0 + 1
= 1
We note that A = 0 since there is
no greater than Fred’s score of 45

i.1
Since two students, Jay and Rod, got the same score of 40,
applying Formula 3.14 for tied scores, we get Rank of Jay &
Rod
=A+
=
Where A= 2 since there are two scores greater than 40 and
B= 2 since there are two students (jay and Rod ) in the knot.

i.2
Next is Ben who obtained a score of 39. By formula 1, Rank of
Ben = A + 1= 4+1 =5, where A= 4 since there are four scores
greater than his score. It follows than rank of Ted is 6. For Ariel,
Mia and Olga, since score of 35, their rank is determined as
follows:
Rank of Ariel, Mia & Olga =
=
Finally, Dick is ranked 10th
, that is, Rank of Dick = A+1=9+1=10,
where A=9 since there are 9 score greater than his score.
Summing up, the ranks of the 10 students on the basis of there
are as follows:

Final Ranking of the Students
Students Scores Ranking
Fred 45 1
Ella 42 2
Jay 40 3.5
Rod 40 3.5
Ben 39 5
Ted 38 6
Ariel 35 8
Mia 35 8
Olga 35 8
Dick 32 10
This table shows the final
ranking of the ten graduate
students who took the 50-item
proficiency test. Summing up
the ranks of the 10 students
on the basis of their scores are
as shown in the table.

2. Fractiles or Quantiles
These measures of position describe or locate the position of a score
or any observation in given data set relative to the entire set of
data. They are value below which a relative number or percentage
of observation in the population must fall. These measure include
the following:
? 2.1 Quartiles. These are Value (denoted by Q1, Q2 , Q3) that
divide a set of data into 4 equal parts such that 25% of the data
falls below Q1, 50% falls below Q2 and 75% falls below Q3.
? 2.2 Deciles. These are Value (denoted by D1, D2, D3) that divide a
set of data into 10 equal parts such that 10% of the data falls
below the D1, 20%, falls below the D2,…., 905 falls below the D3.
? 2.3 Percentiles. These are value (Denoted by P1, P2,……, P99) that
divide a set of data into 100 equal parts such that 1%of the data
falls below P-1, 2% falls below the P2 ,…, and 99% falls below P99.
Based on their definition, it
can be seen that the
following equality relations
between the various fractile
values hold
Median = Q2 = D5 = P50
Q1= P25
Q3= P75
D1= P10
D2= P100
:
D9= P90

3.5 Skewness and Kurtosis
? Large sets of data may also be described in term of its shape and configuration
through measures of skewness and kurtosis. A distribution is said to be symmetric if it
can be folded along a vertical axis so that is two sides coincide (Walpole, 2000). A
distribution that lacks symmetry is said to be asymmetric or skewed thus, symmetry
implies balance in the shape or distribution of a set of measurements. On the other
hand, a distribution on that is skewed to the right (that, it has a long right tail
compared to a much shorter tail) is positively skewed while a distribution that is
skewed to the left is negatively skewed.
Pearsonian Coefficient of Skewness:
SK =

i.
? As shown in formula 3.15, the numerator of SK is three times the
difference of the mean and the median. For a symmetrical
distribution, the mean = median, hence, SK=0. When the distribution
is skewed to the left, SK is negative and when the distribution is
skewed to the right SK is positive. In general, the values of SK will fall
between -3 and 3.
? The distribution in normal, kurtosis = 3. If kurtosis is less than three, the
distribution is platykurtic or less peaked than the normal and a
kurtosis greater than three implies that the distribution is leptokurtic
or more peaked than the normal curve.

Shapes of frequency Distribution
? The measures of skewness and kurtosis are useful in describing the shape of large
sets of data since they indicate the extent of departure of a distribution from
normality and permit comparison of two or more distributions.
? The graphs below present various shapes of distribution described in terms of
central tendency, symmetry and skewness
?
Median
mode
Figure 3.1 A unimodal, Symmetrical Distribution

The Positive and Negative Distribution
A Negatively Skewed Distribution A Positively Skewed Distribution

Descriptive Measures for Grouped Data

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