Business Statistics: Module 2. Descriptive Statistics
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Module 2. Descriptive Statistics: Numerical Characteristics Measures
Descriptive Statistics

tools used to summarize and present the gathered data in graphical, tabular, or
numerical form so readers can easily understand it; its numerical measures are
measures of central tendency, variability, and distribution of shapes.

In this module, we will focus on the ungrouped data and numerical characteristics of
a sample.
Measures of Location/Central Tendency
Central tendency – measures how the data are gathered around the central value

Mean – most common measure of central tendency; average value of a given set of
data; serves as the fulcrum or balance point in a set of data; describes what is
typical in a set of data; all values pay an equal role; its greatly affected by extremely
high or extremely low values.
Mean (x bar) = Ʃx
n
Ʃ = sum of or total
x = a value
n = no. of sample or observations
A survey among randomly select 10 students was conducted to determine their daily
allowances (in pesos). Compute the average daily allowance of the students using
the gathered data shown below
100
150
150
100
200
70
140
170
90
120
First thing to do is add all the values and then divide it by the number of samples
X bar = 100 + 150 + 150 + 100 + 200 + 70 + 140 + 170 + 90 + 120 = 1290 = 129
10
10
We can say that the average daily allowance of the 10 students was Php129

Median – middle value in a given set of data, arranged from lowest to highest; .it is
not affected by extremely high or low values provided there is no change on the
number of data
Using the given data on daily allowances, determine the median daily allowance
100
150
150
100
200
70
140
170
90
120
First thing you need to do is arrange the value from lowest to highest
70
90
100
100
120
140
150
150
170
Then determine the median rank = (n + 1) = (10 + 1) = 11 = 5.5 = 130
2
2
2
200
Business Statistics: Module 2. Descriptive Statistics
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Get the average of the ranks 5 and 6 values= (120 + 140) = 260 = 130
2
2
The median daily allowance of the 10 students was Php130

Mode – simplest measure of central tendency; the value which occur most frequency
in a set of data;
No mode – no value was repeated in a set of data; if there is no repeated value, DO
NOT write 0 nor leave the item blank, write None or No Mode
Unimodal – one value occurs more frequently in a set of data
Bimodal – two modes are in a set of data
Multimodal – more than two modes are in a set of data
The data on daily allowances of the 10 students has two modes, 100 and 150
because both appeared twice in the set of data.
Measures of Variation/Dispersion
Variation – measures how the values are scattered or dispersed from the central
average

Range – also called as spread; simplest measure of variation; represents the
distance or difference of the highest and lowest values in a set of data
Range (R) = Highest value (H) – Lowest value (L)
Using the data given in measures of central tendency, let us measure the variation.
A survey among randomly select 10 students was conducted to determine their daily
allowances (in pesos). Compute for the range.
100
150
150
100
200
70
140
170
90
120
Range = 200 – 70 = 130
The daily allowances of the 10 students were spread by P130.

Interquartile range (IQR) – known as mid-spread; measures the dispersion between
the third and first quartiles
Interquartile range (IQR) = third quartile (Q3) – first quartile (Q1)
Before we can calculate the IQR, we need to determine the third and first quartiles
ranks and values; and the formula are:
Q3 = 3(n+1)
4
Q1 = (n+1)
4
n = number of values or sample in a set of data
Business Statistics: Module 2. Descriptive Statistics
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We also need to arrange the data from lowest to highest. Using the data on the 10
students’ daily allowances, let us arrange it accordingly; and determine the
interquartile range.
70
90
100
100
120
140
150
150
170
200
Q3 = 3(10+1) = 8.25 rank to know the Q3 value, get the 8th value (150) and add it to
4
the product of 0.25 and the difference between 9th (170)
and 8th (150) values
Q3 = 150 + 0.25(170-150) = 155 Q3 value
Q1 = (10+1) = 2.75 rank
4
to know the Q1 value, get the 2nd value (90) and add it to
the product of 0.75 and the difference between 3 rd (100)
and 2nd (90) values
Q1 = 90 + 0.75(100-90) = 97.5 Q1 value
IQR = Q3 – Q1 = 155 – 97.5 = 57.5
The interquartile range of the students’ daily allowance was P57.5.

Variance (S2) – sum of the squared difference of x-value and the mean and divide it
by number of samples minus 1.
S2 = Ʃ(x – xbar)2
(n-1)
x = any value in a set of data
xbar = mean
Ʃ = total or summation
n = number of samples
Let’s compute the variance using the data on students’ daily allowance
x
70
90
100
100
120
140
150
150
170
200
Ʃ 1,290
xbar =
1290 = 129
10

(x – xbar)2
( 70 – 129)2 = 3481
( 90 – 129)2 = 1521
(100 – 129)2 =
841
(100 – 129)2 =
841
2
(120 – 129) =
81
(140 – 129)2 =
121
(150 – 129)2 =
441
(150 – 129)2 =
441
(170 – 129)2 = 1681
(200 – 129)2 = 5041
Ʃ 14490
2
S = 14490 = 1610
10 – 1
First, get the total of the all x values
Next, get the mean or xbar by dividing
total (1290) and number of values
(in this example n is10).
Get the squared difference of each
x value and xbar and then get its total
Next, divide total (14490) and no. of
values – 1 (10 – 1)
The quotient or result of the division is
the variance. (1610)
Standard Deviation (S) – the square root of the variance.
Business Statistics: Module 2. Descriptive Statistics
S=
S2
=
1610
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= 40.12
The daily allowances of the 10 students deviates by P40.12 around the mean of
P129.

Coefficient of Variation (CV) – measure of variation expressed in percentage;
measure the deviation compared with the mean
CV =
S X 100
xbar
%
CV = 40.12 X 100 % = 31.10%
129
The standard deviation (40.12) is 31.10% relative to the mean (129) of the daily
allowances of the 10 students

Z-score – also called standardized value; measures the location of a value in relation
to its deviation from the mean; it identifies extreme value or outlier (value which is
distant from the mean; z-scores beyond + 3.00 are considered outliers.
Z = (x – xbar)
S
Let’s use the lowest and highest daily allowance data and
calculate the lowest and higher z-scores in the said set of data
Zlowest = (70 – 129) = -1.47
40.12
Zhighest = (200 – 129) = 1.77
40.12
Based on the computed z-cores, none of the
values in the set of data is an outlier.
Pattern of Distribution
Shape – shows the pattern of distribution of a set of data, which can either be
symmetrical or skewed

Symmetrical – values are normally distributed; appearance of high and low values in
a set of data are relatively equal
 most of the values are in the middle or near the mean and few high or low values
 bell-shaped
 mean and median values are equal
 zero skewness

Skewed – imbalance/distorted distribution of values, either skewed to left or right

Left skewed or negatively skewed – most of the values are high and some very
low values appear in the set of data cause the distortion to the left tail of the
shape; mean (xbar) is less than (<) median
Business Statistics: Module 2. Descriptive Statistics

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Right skewed or positively skewed – most of the values are low and some very
high values appear in a set of data cause the skewness to the right tail of the
shape; mean (xbar) is greater than (>) median
Let’s determine the shape of data on daily allowance.
The mean and median daily allowances of the 10 students were P129 and P130
respectively. As such, we can say that the data were negatively skewed because
the mean of P129 is less than the median of P130.
End of Module Questions
1. In your opinion, which one among the three central tendency measures is the best to
use to analyze a set of data? Explain why you chose that particular measure.
2. Choose one best measure of variation to use; and explain why you prefer that
particular one over the other measures.
End of Module Exercises
1. The following values were from sample of seven (7): 10, 12, 15, 8, 10, 13, 9
a. Compute the mean, median, and mode.
b. Compute the range, interquartile range, variance, standard deviation, coefficient
of variation.
c. Determine if there any outliers in the set of data.
d. How are the set of data distributed? Present your basis.
2. Analyze the data taken from sample of 15:
3.5; 3.0; 3.75; 4.25; 4.0; 4.88; 3.25; 3.5; 4.0; 3.75; 3.98; 4.15; 4.0; 3.0; 4.0
a. Compute the mean, median, and mode.
b. Compute the range, interquartile range, variance, standard deviation, coefficient
of variation.
c. Determine if there any outliers in the set of data.
d. How are the set of data distributed? Present your basis.
3. A grade 7 student has been saving money every week for the last two months. Her
weekly savings vary depending on her weekly school expenses. The amount of her
weekly savings (in pesos) are as follows: 50, 60, 55, 40, 45, 70, 100, 80.
a. Compute the mean, median, and mode.
b. Compute the range, interquartile range, variance, standard deviation, coefficient
of variation.
c. Determine if there any outliers in the set of data.
d. How are the set of data distributed? Present your basis.
4. A survey was conducted among 12 college students about the number of hours they
spend every day on Facebook and playing online games. The results are shown on
the table:
Business Statistics: Module 2. Descriptive Statistics
Facebook
Online games
3.5
4.5
3
4
2.5
3.5
4
5
4.5
4
2
3.5
1.5
4
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3
4.5
2
3
2.5
3.5
3
2.5
3.5
3
As to students’ numbers of hours spend on facebook and online games, separately:
a. Compute the mean, median, and mode.
b. Compute the range, interquartile range, variance, standard deviation, coefficient
of variation.
c. Determine if there any outliers in the set of data.
d. How are the set of data distributed? Present your basis.
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https://www.youtube.com/watch?v=mk8tOD0t8M0.
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