DRIVE SPRING DRIVE 2015
PROGRAM BACHELOR OF BUSINESSADMINISTRATION (BBA)
SUBJECT CODE & NAME BBA104-Quantitative Technique inBusiness
BK ID B1500
1. A measure of central tendency is a single value that attempts to describe a set of data by
identifying the central position within that set of data. As such, measures of central
tendency are sometimes called measures of central location. They are also classed as
summary statistics. The mean (often called the average) is most likely the measure of
central tendency that you are most familiar with, but there are others, such as the median
and the mode.
The mean, median and mode are all valid measures of central tendency, but under
different conditions, some measures of central tendency become more appropriate to use
A mathematical representation of the typical value of a series of numbers,
computed as the sum of all the numbers in the series divided by the count of all numbers
in the series. Arithmetic mean is commonly referred to as "average" or simply as
"mean". The arithmetic mean, also called the average or average value, is the quantity
obtained by summing two or more numbers or variables and then dividing by the
number of numbers or variables. The arithmetic mean is important in statistics. When
there are only two quantities involved, the arithmetic mean is obtained simply by
adding the quantities and dividing by 2. In these cases, the operation is sometimes
symbolized by a double colon (::) between the two quantities to be averaged. For
example: 3 :: 11 = 7-10 :: +4 = -3
2. The determination of the average of a large number of quantities is a tedious
task; computers are commonly used to calculate these values. The arithmetic mean of a
continuous function over a defined interval is determined by first calculating the definite
integral over the interval, and then dividing this quantity by the width of the interval.
The average of a set of products, the calculation of which is commonly used to
determine the performance results of an investment or portfolio. Technically defined as
"the 'n'th root product of 'n' numbers", the formula for calculating geometric mean is most
easily written as:
Where 'n' represents the number of returns in the series.
The geometric mean must be used when working with percentages (which are derived
from values), whereas the standard arithmetic mean will work with the values
The median is defined as the number in the middle of a given set of numbers
arranged in order of increasing magnitude. When given a set of numbers, the median is
the number positioned in the exact middle of the list when you arrange the numbers from
the lowest to the highest. The median is also a measure of average. In higher level
statistics, median is used as a measure of dispersion. The median is important because it
describes the behavior of the entire set of numbers.
The mode is defined as the element that appears most frequently in a given set of
elements. Using the definition of frequency given above, mode can also be defined as the
element with the largest frequency in a given data set. For a given data set, there can be
3. more than one mode. As long as those elements all have the same frequency and that
frequency is the highest, they are all the modal elements of the data set.
2. Cumulative frequency distribution:
The total frequency of all classes less than the upper class boundary of a given
class is called the cumulative frequency of that class. A table showing the cumulative
frequencies is called cumulative frequency distribution. There are two types of
cumulative frequency distributions.
a) Less than cumulative frequency distribution: It is obtained by adding successively
the frequencies of all the previous classes including the class against which it is written.
The cumulate is started from the lowest to the highest size.
b) More than cumulative frequency distribution: It is obtained by finding the
cumulate total of frequencies starting from the highest to the lowest class.
One of the most common ways to represent data graphically is called a pie chart. It
gets its name by how it looks, just like a circular pie that has been cut into several slices.
This kind of graph is helpful when graphing qualitative data, where the information
describes a trait or attribute and is not numerical. Each trait corresponds to a different
slice of the pie. By looking at all of the pie pieces, you can compare how much of the
data fits in each category. The larger the category the
bigger that its pie piece will be.
How do we know how large to make a pie
piece? First we need to calculate a percentage. Ask
what percent of the data is represented by a given
category. Divide the number of elements in this
category by the total number. We then convert this
decimal into a percentage.
4. A pie is a circle. Our pie piece, representing a given category, is a portion of the circle.
Because a circle has 360 degrees all the way around, we need to multiply 360 by our
percentage. This gives us the measure of the angle that our pie piece should have.
A bar graph displays data visually and is sometimes called a bar chart or a
bar graph. Data is displayed either horizontally or vertically and allows viewers to
compare items displayed. Data displayed will relate to things like amounts,
characteristics, times and frequency etc. A bar graph displays information in a way that
helps us to make generalizations and conclusions quickly and easily.
A typical bar graph will have a label, axis, scales and bars. Bar graphs are used to
display all kinds of information such as, numbers of females versus males in a school,
sales of items during particular times of a year. Bar graphs are ideal for comparing two or
The image shows a bar graph about the favorite seasons of the boys and girls in a
class. The blue represents the fall, the red represents the winter, the green represents the
spring and the purple represents the fall. At a glance, you can see that the favorite season
among the girls is summer and the boy is fall. The least favorite season for both girls and
boys is winter. On the left side of the graph are numbers that represent how many
students were surveyed. Numbers of students are represented on the x axis and the y axis
shows the genders.
When interpreting a bar chart, look at the tallest bar and look at the shortest bar. Look at
the titles, look for inconsistencies and ask why they are there.
Types of Bar Graphs:
Single bar graphs are used to convey the discrete value of the item for each
category shown on the opposing axis.
An example would be a representation of the number of males in grades 4-6 for
each of the years 1995 - 2010. The actual number (discrete value) could be represented
5. by a bar sized to scale with the scale appearing on the x axis. The Y axis would show a
tick and label for the corresponding year for each bar.
A grouped or clustered bar graph is used to represent discrete values for more than
one item that share the same category. An example would be, using the single
bar example above and introduce the number of female students in grades 4-6 for the
same categories, years 1995- 2010. The two bars would be grouped together, side by side,
and each could be color coded to make it clear which bar represents male vs. female
Some bar graphs have the bar divided into subparts that represent the discrete value for
items that represent a portion of a whole group. An example would be to represent the
actual grade data for males in each grade 4-6 and then scale each grade discrete value as a
part of the whole for each bar. Again color coding would be needed to make the graph
A histogram is a type of graph that has wide applications in statistics. Histograms
allow a visual interpretation of numerical data by indicating the number of data points
that lie within a range of values, called a class or a bin. The frequency of the data that
falls in each class is depicted by the use of a bar.
Histograms are used for data that is at least at the ordinal level of measurement.
The classes for a histogram are ranges of values. The bars in a histogram cannot be
rearranged. They must be displayed in the order that the classes occur.
A frequency polygon is a graph showing the differences in frequencies or
percentages among categories of an interval-ratio variable. Points representing the
frequencies of each category are placed above the midpoint of the category and are joined
by a straight line. A frequency polygon is similar to a histogram, however instead of bars,
a point is used to show the frequency and all the points are then connected with a line.
6. 3. Trend:
A pattern of gradual change in a condition, output, or process, or an average or
general tendency of a series of data points to move in a certain direction overtime,
represented by a line or curve on a graph.
Method of determining trend in time series:
This method is most widely used in practice. It is mathematical method and with
its help a trend line is fitted to the data in such a manner that the following two conditions
1. 0)( cYY i.e. the sum of the deviations of the actual values of Y and the
computed values of Y is zero.
)( cYY is least, i.e. the sum of the squares of the deviations of the actual values
and the computed values is least.
The line obtained by this method is called as the “line of best fit”. This method of least
squares may be used either to fit a straight line trend or a parabolic trend.
Fitting of a straight line trend by the method of least squares:
Let tY be the value of the time series at time t. Thus tY is the independent variable
depending on t.
Assume a straight line trend to be of the form btaYtc …………. (1)
Where tcY is used to designate the trend values to distinguish from the actual tY values, a
is the Y-intercept and b is the slope of the trend line.
Now the values of a and b to be estimated from the given time series data by the method of
In this method we have to find out a and b values such that the sum of the squares of
the deviations of the actual values tY and the computed values tcY is least.
)( tct YYS should be least
)( btaYS t ………… (2) Should be least
Now differentiating partially (2) w.r.to a and equating to zero we get
tbnaYt ……………….. (3)
Now differentiating partially (2) w.r.to b and equating to zero we get
The equations (3) and (4) are called ‘normal equations’
Solving these two equations we get the values of a and b say aˆ andbˆ .
Now putting these two values in the equation (1) we get
This is the required straight line trend equation.
Fitting the straight line trend
X XY X2 Trend Values
2000 4 -2 -8 4 4.4
2001 6 -1 -6 1 6.2
2002 9 0 0 0 8
2003 10 +1 +10 1 9.8
2004 11 +2 +22 4 11.6
N = 5 ∑Y = 40 ∑X = 0 ∑XY = 18 ∑X2 = 10 ∑YC = 40
8. The equation of a straight line is YC = a + bX
To find a and b we have two normal equations
N = 5, ∑Y = 40, ∑X = 0, ∑XY = 18, ∑X2 = 10
∑X = 0; a = ∑Y/N = 40/5 = 8
b = ∑XY/∑X2 = 18/10 = 1.8
Hence, the equation of a straight line trend is YC = 8 + 1.8X
For X = -2, YC = 8 + 1.8(-2) = 4.4
For X = -1, YC = 8 + 1.8(-1) = 6.2
For X = 0, YC = 8 + 1.8(0) = 8
For X = 1, YC = 8 + 1.8(1) = 9.8
For X = 2, YC = 8 + 1.8(2) = 11.6