Data Analysis

Data can be organised in a number of ways so that larger volume of data can be presented in a more compact and precise form.

Data thus presented has to be deciphered correctly by the user of the data.

This process of deciphering the data from its compactly presented form is called Data Interpretation.

Methods of Presenting Data:

Numerical data can be presented in one or more of the following ways:

1. Data Tables

2. Pie Chart

3. Bar Chart

4. Line Graph

5. Function

1) Data Tables

Data is presented in the form of simple table into rows and columns.

Example 1-4:

Study the table shown below and answer the questions.

Table shows number of candidates appeared, qualified and selected in a CSIR NET examination from five states - Delhi, H.P, U.P, Punjab and Haryana over the Years 1997 to 2001

Example 1: Average selection of which state is maximum?

The average number of candidates selected over the given period for various states are:

Delhi = (94+48+82+90+70)/5=384/5=76.8

H.P. = (82+65+70+86+75)/5=378/5=75.6

U.P. = (78+85+48+70+80)/5=361/5=72.2

Punjab = (85+70+65+84+60)/5=364/5=72.8

Haryana = (75+75+55+60+75)/5=340/5=68

It's clear that average selection is maximum for Delhi.

Example 2: The percentage of candidates qualified from Punjab over those appeared from Punjab is highest in the year?

The percentages of candidates qualified from Punjab over those appeared from Punjab during different years are:

For year 1997 = (680/8200*100)%=8.29%

For year 1998 = (600/6800*100)%=8.82%

For year 1999 = (525/6500*100)%=8.08%

For year 2000 = (720/7800*100)%=9.23%

For year 2001 = (485/5700*100)%=8.51%

So, the percentage of candidates qualified from Punjab is highest for the year 2000.

Example 3: In the year 1997, which state had the lowest percentage of candidates selected over the candidates appeared?

The percentages of candidates selected over the candidates appeared in 1997, for various states are:

For Delhi = (94/8000*100)%=1.175%

For H.P. = (82/7800*100)%=1.051%

For U.P. = (78/7500*100)%=1.040%

For Punjab = (85/8200*100)%=1.037%

For Haryana = (75/6400*100)%=1.172%

So, the lowest percentage of candidates selected over the candidates appeared for Punjab.

Example 4: The number of candidates selected from Haryana during the period under review is approximately what percent of the number selected from Delhi during this period?

Required Percentage = [(75+75+55+60+75)/(94+48+82+90+70 )*100]%

= [340/(384 )*100]%

= 88.54 % = 88.5%

2) Pie Chart

A pie chart is a circular chart in which data is displayed in the form of pie slices.

The size of an arc of each slice is proportional to the quantity it represents.

The bigger the slice, the bigger portion of the whole is represented by it.

The total angle formed at the centre is 360 degrees, which is the sum of the fractions of all the sectors.

A pie chart basically shows the relationship between the whole and its parts. Thus, for its construction, we need to find how they are related as a fraction, and then convert it into a scale of 360 degrees to obtain the angle that the part subtends at the center.

** Angle of a component = (Value of the component)/(Total Value) x 360 ^{0} **

Example 5 - 8:

The following pie-chart shows the percentage distribution of the expenditure incurred in publishing a book.

Study the pie-chart and the answer the questions based on it.

Example 5: If for a certain quantity of books, the publisher has to pay Rs. 30,600 as printing cost, then what will be amount of royalty to be paid for these books?

Let the amount of Royalty to be paid for these books be Rs. x.

Then, 20: 15 = 30600: x

x = Rs. (30600 x 15)

= Rs. 22,950

Example 6: What is the central angle of the sector corresponding to the expenditure incurred on Royalty?

Central angle corresponding to Royalty = (15% of 360)^{0}

= (15/100 x 360)^{0}

= 540

Example 7: The price of the book is marked 20% above the C.P. If the marked price of the book is Rs. 180, then what is the cost of the paper used in a single copy of the book?

Marked price of the book = 120% of C.P.

Cost of paper = 25% of C.P

Let the cost of paper for a single book be Rs. n.

Then, 120: 25 = 180: n

n = Rs.25 x 180= Rs 37.50

Example 8: If 5500 copies are published and the transportation cost on them amounts to Rs. 82500, then what should be the selling price of the book so that the publisher can earn a profit of 25%?

For the publisher to earn a profit of 25%, S.P. = 125% of C.P.

Also Transportation Cost = 10% of C.P.

Let the S.P. of 5500 books be Rs. x.

Then, 10 : 125 = 82500 : x

x = Rs.125 x 8250= Rs. 1031250.

S.P. of one book = Rs.1031250/5500

= Rs. 187.50

3) Bar Chart

A bar chart portrays a visual interpretation of data with the help of vertical or horizontal rectangular bars where the lengths of the bars are proportional to the data to be represented.

Example 9: In a school of 400 students, the percentage of attendance of students is represented by the following table.

By observing the bar graph it can be concluded that:

The number of students with 60% attendance is 105,

The number of students with 70% attendance is 199,

The number of students with 80% attendance is 29 and

The number of students with 90% attendance is 73.

4) Line Graph

A line graph displays information as a series of data points called 'markers' connected by straight line segments.

Line graphs display data in two dimensions - x-axis and the y-axis.

The dependent variable or y variable is on the vertical axis and the independent or x variable is on the horizontal axis.

The rise and falls in the line shows how one variable is affected by the other.

Example 10 - 12:

The following line graph gives the percentage of the number of candidates who qualified an examination out of the total number of candidates who appeared for the examination over a period of seven years from 1994 to 2000.

Example 10: The difference between the percentages of candidates qualified to appear was maximum in which of the following pairs of years?

The differences between the percentages of candidates qualified to appear for the give pairs of years are:

For 1994 and 1995 = 50 - 30 = 20

For 1998 and 1999 = 80 - 80 = 0

For 1994 and 1997 = 50 - 30 = 20

For 1997 and 1998 = 80 - 50 = 30

For 1999 and 2000 = 80 - 60 = 20

Thus, the maximum difference is between the years 1997 and 1998.

Example 11: In which pair of years was the number of candidates qualified the same?

The graph gives the data for the percentage of candidates qualified to appear and unless the absolute values of number of candidates qualified or candidates appeared is know we cannot compare the absolute values for any two years.

Hence, the data is inadequate to solve this question.

Example 12: If the total number of candidates appeared in 1996 and 1997 together was 47400, then the total number of candidates qualified in these two years together was?

The total number of candidates qualified in 1996 and 1997 together, cannot be determined until we know at least, the number of candidates appeared in any one of the two years 1996 or 1997 or the percentage of candidates qualified to appeared in 1996 and 1997 together.

Hence, the data is inadequate.

5) Function

Function represents the dependence between variable quantities in the process of their changes.

For Example: With a change in the side if a square, the area of the square also changes.

Function is symbolized as **y = f(x)** where f represents the rule by which y varies with x, where "x" is independent variable and "y" is dependent variable.

Function can be represented by below ways:

(1) Analytical Representation

(2) Tabular Representation

(3) Graphical Representation

1. Analytical Representation:

Here function is represented through a formula.

a) This representation of function could be a uniform formula in the entire domain.

Example: y = 4 x^{2}

b) Or By several formulae which are different part of the domain.

Example: y = 4 x^{2} if x < 0 and y = x^{2} if x > 0

2. Tabular Representation:

For representing functions through table, write down a sequence of values of the independent variable "x" and note down the corresponding value of the dependent variable 'y'.

Tables of trigonometric, logarithms etc. are tabular representation of the functions.

Example:

3. Graphical Representation:

In such representation function is produced in 'xy' coordinate plane for every value of x from the domain D of the function, a point P(x, y) is constructed whose abscissa is x and whose ordinate y is got by putting the particular value of x in the formula representing the function.

Example: y = x + 1

Some generic types of Functions:

(1) Even Function:

Let a function y = f(x) be given in a certain interval. The function is said to be even if for any value of x:

f(x) = f(-x)

Properties of Even Functions:

(a) The sum, difference, product and quotient of an even function is also an even function.

(b) The graph of an even function is symmetric about the y-axis.

Example: y = x^{2}, y = x^{4}

(2) Odd Function:

Let a function y = f(x) be given in a certain interval. The function is said to be odd if for any value of x:

f (-x) = -f(x)

Properties of Even Functions:

(a) The sum and difference of an odd function is also an odd function.

(b) The product and quotient of an odd function is an even function.

Note: Not all functions need be even or odd. However, every function can be represented as the sum of an even function and an odd function.

Example 13: The speed of the truck increases every minute as shown in below table.
What would be the speed of the truck at the end of the 19^{th} minute?

From the above table, it's clear that speed = 1.5 times of time i.e. speed = 1.5 (time).

So the speed after 19th minute = 1.5 x 19 = 28.5 m/sec

Example 14: Which of the following straight line passes through the point (1, 1)?

Put the value of x = 1 in all the options and the equation which satisfy the point (1,1) is the correct option.

Option (1):

y = 2x + 3 = 2 + 3 = 5

Hence, this is wrong option.

Option (2):

2y = x - 6

y = (x - 6)/2=(1- 5)/2= -5/2

Hence, this is wrong option.

Option (3):

x = 1

Here x will remain as 1 for all values of y and hence this is wrong option.

Option (4):

y = x = 1

Hence this is the correct option.

Example 15: A football is dropped from a height h above the surface of the earth. Ignore air drag, the curve that rest represents its variation of acceleration is:

Note: For freely falling body gravity/acceleration must be constant.

Only option (4) represents that acceleration is constant, in all other options value of acceleration is either increasing or decreasing.

Example 16: Suppose y = e^{x}, then graph of this function is represented by:

Given: y = e^{x}

If x = 1, y = e^{1} = 2.718 (Note: Value of e = 2.718)

If x = 2, y = e^{2} = 7.382

Option (2) and (4) is wrong because when x = 0, y value should not be 0 as per given function.

y = e^{x} = e^{0} = 1

So value of y should be 1 when x = 0.

Option (3) is wrong because with increase in value of x , value of y should increase as per given function. Bit in graph, value of y is decreasing with increase in value of x.

Option (1) is correct because here value of y is increasing faster than value of x as per given function.