CHAPTER FIVE

SIM

PLE PERCENTAGE

5.1Introduc

tion

Stati

st

ical data are

obtained thr

ough

measurem

ent or counting. They a

re round off to make the data c

learer and more understandable.

This assists in minimizing erro

rs when large numbers are invol

ved. This will be discussed wi

th simple percentage. Simple

percentage is another form

of descriptive statistics u

sed in analyzing data.

5.2Objectives

At the end of this chapter, you should be able:

  1. round off data

  2. convert data to percentage.

  3. Calculate actual error

  4. Compute percentage error

5.3Rounding off Data

Statistical data collected through measurement or counting are approximated into a desired degree of accuracy, either rounding off upwards or downward. Rounding off is the method of approximating a number to the nearest unit, hundred, thousand, million, decimal places or significant figures.

The digits 1, 2, 3, 4 are rounded down and they are called zero. While the digits 5, 6, 7, 8, 9, are rounded up called 1 and added to the next digit.

Let us illustrate how to round off numbers to the nearest tens, hundreds, or thousands.

Examples

  1. Approximate each of the following to the nearest tens, hundreds, and thousands.

  3. 13657

  4. 76523

  5. 13,657 becomes 13660 the nearest ten, round up

  6. 13657 becomes 13700 the nearest hundred round up

  7. 13657 becomes 14,000 to the nearest thousands round up.

  1. 76523 = 76520 to nearest ten round down

  2. 76523 = 76500 to nearest hundred round down

  3. 76523 = 77000 to nearest thousand round up.

Note that when measurement is made, the approximation or round off makes the data collected clear and more understandable. It also assists, in minimizing cumulative errors when a large number of operations are carried out.

5.4Decimal Point and Significant Figure

Through statistical manipulation, the measurement-based statistical data may be made more understandable or acceptable. For example, they can be rounded to a predetermined number of significant figures or decimal places.

5.4.1Significant Figures

The significant figure, abbreviated as (sf), is calculated starting from the non-zero numeral located to the number’s left. The leftover numbers are removed using the following guidelines once the necessary number of significant figures has been recorded:

If a set of numbers is to be eliminated and the first digit is a 5 or above, the last digit is raised by 1.

For example:

1. Approximate 7.4543 to two, three, significant figures

7.4543 = 7.5 to 2 sf

7.4543 = 7.45 to 3 sf

2. Approximate 0.00758 to one sf, two sf.

  1. 0.00758 the zero before the real number is not a significant figure. The rounding off will take place after the first real number.

0.00758 = 008 to 1 sf

  1. 0.00758 = 0.0076 to 2 sf.

Note that the number zero is only significant if only situated after any non-zero real number in the whole number part e.g., 3406, the zero here is significant, but in 0.069, 15.40 and 0.000056 are zeros that are not significant.

5.4.2Decimal Point

Decimal point is sometimes referred to decimal place and abbreviated to dp. These are counted to the Right of the decimal point and contained the same rules of rounding off in significant figures. For example,

Round off each of the numbers to

  1. one decimal place

  2. two decimal places

  3. 0.005

  4. 6.5020

  5. 0.005 = 0.0 to 1 dp

= 0.01 to 2 dp

  1. 6.5020 = 6.5 to 1 dp

= 6.50 to 2 dp

5.5Percentage

Percentage is a useful statistic used for describing information obtained. It is good in describing the features of two or more groups of people or objects where the number of people or objects in the groups are different. At this particular period, reporting only the frequency of such features is not enough. But if the frequency of people in each group who possess the relevant features is converted to the percentage, then the group can be compared. Because the percentage treats the groups as though they are of the same size (i.e., 100).

Simple percentage is given as = \\frac{\\mathrm{N}\\mathrm{u}\\mathrm{m}\\mathrm{b}\\mathrm{e}\\mathrm{r}\\mathrm{o}\\mathrm{f}\\mathrm{F}\\mathrm{r}\\mathrm{e}\\mathrm{q}\\mathrm{u}\\mathrm{e}\\mathrm{n}\\mathrm{c}\\mathrm{i}\\mathrm{e}\\mathrm{s}}{\\mathrm{T}\\mathrm{o}\\mathrm{t}\\mathrm{a}\\mathrm{l}\\mathrm{n}\\mathrm{u}\\mathrm{m}\\mathrm{b}\\mathrm{e}\\mathrm{r}\\mathrm{o}\\mathrm{f}\\mathrm{f}\\mathrm{r}\\mathrm{e}\\mathrm{q}\\mathrm{u}\\mathrm{e}\\mathrm{n}\\mathrm{c}\\mathrm{i}\\mathrm{e}\\mathrm{s}\\mathrm{i}\\mathrm{n}\\mathrm{t}\\mathrm{h}\\mathrm{e}\\mathrm{g}\\mathrm{r}\\mathrm{o}\\mathrm{u}\\mathrm{p}}\\times 100

A Researcher obtained data in School of Arts and Social Sciences and is interested in comparing the performance of students in the different department in the School as shown in Table 5.1.

Table 5.1: Shows 2020/2021 session number of Students admitted and those who passed without any carry over.

Department

No. of Admitted

No. Passed

Economics

850

400

Geography

250

200

History

180

150

Social Studies

900

300

The percentage of Students who passed without any carry over in each department of School of Arts and Sciences is as follows:

  1. Percentage who passed Economics=\\frac{400}{850}\\times \\frac{100}{1}\\times =47.05^{\\circ}\\%

  2. Percentage who passed Geography=\\frac{200}{250}\\times 100=80^{\\circ}\\%

  3. Percentage who passed History=\\frac{150}{180}\\times 100=83.33^{\\circ}%

  4. Percentage who passed S/studies=\\frac{300}{900}\\times 100=33.33^{\\circ}\\%

If you compare the number of frequency of Students who passed in Economics department without any carry over, which is (400) and those in the Geography department that is you may be tempted to conclude that performance in Economics is better than Geography. But if you use the percentage values you may see that performance in Geography is better than performance in Economics.

5.5.1Percent Error

Percent Error is degree or level of difference between estimate value and the actual value in comparing to the actual value and is expressed as a researcher on taking decision on the acceptable level of error. The error may be either positive or negative.

Let us look into this example. A student in a lab measure the lenght of alminium glass and accidentally records 10 M if the actual lenght is 12 M

To calculate the percentage error, the followings steps shall be adopted.

  1. Subtract the actual value from the estimate value

  2. Divide the result obtained from step I

  3. Multiply the result by 100 to obtain total percentage.

  4. Add percent or % symbol to report your result (i.e. percent error values)

To solve for this example, you use the formula

Percentage error (P.E) =\\frac{\\mathrm{E}\\mathrm{s}\\mathrm{t}\\mathrm{i}\\mathrm{m}\\mathrm{a}\\mathrm{t}\\mathrm{e}\\mathrm{d}\\mathrm{V}\\mathrm{a}\\mathrm{l}\\mathrm{u}\\mathrm{e}\\left(\\mathrm{e}\\mathrm{V}\\right)-\\mathrm{A}\\mathrm{c}\\mathrm{t}\\mathrm{u}\\mathrm{a}\\mathrm{l}\\mathrm{V}\\mathrm{a}\\mathrm{l}\\mathrm{u}\\mathrm{e}(\\mathrm{A}\\mathrm{V})}{\\mathrm{A}\\mathrm{c}\\mathrm{t}\\mathrm{u}\\mathrm{a}\\mathrm{l}\\mathrm{V}\\mathrm{a}\\mathrm{l}\\mathrm{u}\\mathrm{e}\\left(\\mathrm{A}\\mathrm{V}\\right)}\\times 100

\\boldsymbol{S}\\boldsymbol{o}\\boldsymbol{l}\\boldsymbol{u}\\boldsymbol{t}\\boldsymbol{i}\\boldsymbol{o}\\boldsymbol{n}

Actual value = 12 M

Estimated Value = 10 M

Step I:10\\mathrm{M}\\hbox{--}12\\mathrm{M}=\\hbox{--}2\\mathrm{M}

Step II: Divide the result with actual values - \\frac{\\hbox{--}2\\mathrm{M}}{12\\mathrm{M}}=\\hbox{--}0.17M

Step III: Multiply the result by 100 to obtain total percentage 0.17 \\times 100 = 17

Step IV: Add percent –17%.

\\boldsymbol{E}\\boldsymbol{x}\\boldsymbol{a}\\boldsymbol{m}\\boldsymbol{p}\\boldsymbol{l}\\boldsymbol{e}

A table of lenght 25cm was measured by a male student to be 24.6cm. Find the percentage error.

The actual error is 25—24.6 = 0.4cm

Percentage error = \\frac{0.4}{25}\\times 100=0.016\\times 100

=1.6\\mathrm{\\% }

\\boldsymbol{E}\\boldsymbol{x}\\boldsymbol{a}\\boldsymbol{m}\\boldsymbol{p}\\boldsymbol{l}\\boldsymbol{e}

A block which weighs 30.5kg is obtained to have weighed 32.6kg. Find the percentage error.

\\boldsymbol{S}\\boldsymbol{o}\\boldsymbol{l}\\boldsymbol{u}\\boldsymbol{t}\\boldsymbol{i}\\boldsymbol{o}\\boldsymbol{n}

The actual error = 32.6—30.5 = 2.1kg

The percentage error = =\\frac{2.1}{30.5}\\times 100=6.9\\%

Student Activity

  1. What is rounding off data?

  2. What is benefit of rounding off data?

  3. Round off each of the following numbers into nearest unit, ten, hundred and thousand.

  4. 24567

  5. 88732

  1. Round off the following numbers into

  2. 1 dp

  3. 3 dp

  4. 1 sf

  5. 3 sf

  6. 00.7568 b. 10.768 c. 0.0759

  7. The number of students in different levels who are intelligent and the total number of students in such levels are given bellow:

TOTAL NUMBER OF STUDENTS

NUMBER OF INTELLIGENT STUDENTS

100LEVEL

30

6

200LEVEL

25

2

300LEVEL

20

4

What percentage of students that are gifted in each level?

  1. In research to compare academic withdraw rates in department of chemistry in three levels. A researcher obtained the following data.

Numbers of Admitted

Number of Withdrawn

300LEVEL

80

40

200LEVEL

30

10

100LEVEL

50

20

  1. What is the percentage withdrawing rate in each of the three levels?

  2. What is the percentage of the total number of the student in the three-level withdrawn?

  3. A researcher was to subtract 10 from a certain number but mistakenly added 15 and obtain the result 140. Determine the percentage error during the data collection.

  4. The student in Mathematics laboratory measured the length of electrical wire as 16.55, while the length is 16.25. Calculate the percentage error and correct the result to 1 decimal place.

  5. If the age of my wife is 45 years is recorded as 61 years, calculate the percentage error to 2 significant figures.

References

National Teachers’ Institute (2000). NCE/DLS Course Book on Education.

Razaq D. & Ajayi, O.O (ND) Research Methods and Statistical Analysis.