## Rounding off

In most of the everyday situations, we do not need to use highly sensitive measuring devices (instruments). the accuracy of our measurement depends on the purpose for which we use the information.

Example: Suppose someone uses a compass as a guide in going from one end of the school to the other. It would not be a serious error if he/ she is 1o off course. However, 1o off course on a journey to the moon will mean an error of 644000 km.

Besides the error arising from the use of different instrument/ devices, the person taking the measurement is another source of error. For example, in school/ college athletics meet, there are usually two or more time-keepers for the first placing of a (say) 100-meter race, and time-keepers may have slightly different time on their devices (such as sports watch). Therefore, all physical measurements such as mass, length, time, volume and area can never be absolutely accurate. The accuracy depends on the degree of the measuring device (instrument) and the person recording (taking) the measurement. Both of them can never be absolutely accurate.

### Rules for Rounding off Numbers

Rule 1: Determine what your rounding digit is and look at the digit to the right of it. If the number is 1, 2, 3, or 4, simply drop all digits to the right of rounding digit. For example,

5.432 may be rounded off to 5.42 nearest to the hundredths place.
5.432 may be rounded off to 5.4 nearest to the tenths place.
5.432 may be rounded off to 5 nearest to units place.

Rule 2: Determine what your rounding digit is and look at the digit to the right of it. If the number is 5, 6, 7, 8 or 9 add one to the rounding digit and drop all digit to the right of rounding digits. For example,

3.786 may be rounded off to 3.79 nearest to the hundredths place.
3.786 may be rounded off to 3.8 nearest to the tenths place.
3.876 may be rounded off to 3.9 nearest to units place.

## Number System

In early civilizations, the number of animals (sheep, goat, and camel etc.) or children people have were tracked by using different methods such as people match the number of animals with the number of stones. Similarly, they count the number of children with the number of notches tied on a string or marks on a piece of wood, leather or wall. With the development of human, other uses for numerals were found and this led to the invention of the number system.

## Natural Numbers

Natural numbers are used to count the number of subjects or objects. Natural numbers are also called counting numbers. The numbers $1, 2, 3, \cdots$

are all natural numbers.

## Whole Numbers

The numbers $0, 1, 2, \cdots$ are called whole numbers. It can be observed that whole numbers except 0 are natural numbers.

## Number Line

Whole numbers can be represented by points on a line called the number line. For this purpose, a straight line is drawn and a point is chosen on the line and labeled as 0. Starting with 0, mark off equal intervals of any suitable length. Marked points are labeled as $1, 2, \cdots$ as shown in Figure below. The figure below represents real numbers since it includes the negative number (numbers on the left of 0 in this diagram are called negative numbers).

The arrow on the extreme (right-hand side in case of while numbers or negative numbers) indicates that the list of numbers continues in the same way indefinitely.

A whole number can be even or odd. An even number is a number which can be divided by 2 without leaving any remainder. The numbers $0, 2, 4, 6, 8, \cdots$ are all even numbers. An odd number is a number which cannot be divided by 2 without leaving any remainders. The numbers $1, 3, 5, 7, 9, \cdots$ are all odd numbers.

It is interesting to know that any two numbers can be added in any order and it will not affect the results. For example, $3+5 = 5+3$. This is called the commutative law of addition. Similarly, the order of grouping the numbers together does not affect the result. For example, $2+3+5=(2+3)+5 = 2+ (3+5)=(2+5)+3$. This is called the associative law of addition. The subtraction and division of numbers are not commutative as $5-7\ne7-5$ and $6\div2 \ne 2\div 6$ in general.

Like addition and multiplication, whole numbers also follow commutative law and it is called commutative law of multiplication, for example, $2\times 8 = 8 \times 2$. Like addition and multiplication, whole numbers also follow associative law of multiplications. For example, $2 \times (3 \times 4) = (2 \times 3) \times 4 = (2 \times 4)\times 3$. Similarly, multiplication is distributive over addition and subtraction, for example, (i) $5\times (6 + 7) = (5 \time 6) + (5 \times 7)$ or $(6+7) \times 5=(6 \times 5)+(7 \times 5)$. (ii) $3 \times (6-2) = (3 \times 6) - (3 \times 2)$ or $(6-2) \times 3 = (6 \times 3) - (2 \times 3)$.

Take any two digit number say 57, reverse the digits to obtain 75. Now subtract the smaller number from the bigger number, we have $75-57=18$. Now reverse the digits of 18 and add 18 to its reverse (81), that is, 18+81, you will get 99.

## Chi-Square Test of Independence

Chi-square test is a non-parametric test. The assumption of normal distribution in the population is not required for this test. The statistical technique chi-square can be used to find the association (dependencies) between sets of two or more categorical variables by comparing how close the observed frequencies are to the expected frequencies. In other words, a chi square ($\chi^2$

) statistic is used to investigate whether the distributions of categorical variables different from one another. Note that the response of categorical variables should be independent from each other. We use the chi-square test for relationship between two nominal scaled variables.

Chi-square test of independence is used as tests of goodness of fit and as tests of independence. In test of goodness of fit, we check whether or not observed frequency distribution is different from the theoretical distribution, while in test of independence we assess, whether paired observations on two variables are independent from each other (from contingency table).

Example: A social scientist sampled 140 people and classified them according to income level and whether or not they played a state lottery in the last month. The sample information is reported below. Is it reasonable to conclude that playing the lottery is related to income level? Use the 0.05 significance level.

 Income Low Middle High Total Played 46 28 21 95 Did not play 14 12 19 45 Total 60 40 40 140

Step by step procedure of testing of hypothesis about association between these two variable is described, below.

Step1:
$H_0$: There is no relationship between income and whether the person played the lottery.
$H_1$: There is relationship between income and whether the person played the lottery.

Step2: Level of Significance 0.05

Step 3: Test statistics (calculations)

 Observed Frequencies ($f_o$$f_o$) Expected Frequencies ($f_e$$f_e$) $\frac{(f_o - f_e)^2}{f_e}$$\frac{(f_o - f_e)^2}{f_e}$ 46 95*60/140= 40.71 $\frac{(46-40.71)^2}{40.71}$$\frac{(46-40.71)^2}{40.71}$ 28 95*40/140= 27.14 $\frac{(28-27.14)^2}{27.14}$$\frac{(28-27.14)^2}{27.14}$ 21 95*40/140= 27.14 $\frac{(21-27.14)^2}{27.14}$$\frac{(21-27.14)^2}{27.14}$ 14 45*60/140= 19.29 $\frac{(14-19.29)^2}{19.29}$$\frac{(14-19.29)^2}{19.29}$ 12 45*40/140= 12.86 $\frac{(12-12.6)^2}{12.86}$$\frac{(12-12.6)^2}{12.86}$ 19 45*40/140= 12.86 $\frac{(19-12.86)^2}{12.86}$$\frac{(19-12.86)^2}{12.86}$ $\chi^2=\sum[\frac{(f_0-f_e)^2}{f_e}]=$$\chi^2=\sum[\frac{(f_0-f_e)^2}{f_e}]=$ 6.544

Step 4: Critical Region:
Tabular Chi-Square value at 0.05 level of significance and $(r-1) \times (c-1)=(2-1)\times(3-1)=2$ is 5.991.

Step 5: Decision
As calculated Chi-Square value is greater than tabular Chi-Square value, we reject $H_0$, which means that there is relationship between income level and playing the lottery.

Note that there are several types of chi-square test (such as Yates, Likelihood ratio, Portmanteau test in time series) available which depends on the way data was collected and also the hypothesis being tested.

## Mean: Measure of Central Tendency

The measure of Central Tendency Mean (also know as average or arithmetic mean) is used to describe the data set as a single number (value) which represents the middle (center) of the data, that is average measure (performance, behaviour, etc) of data. This measure of central tendency is also known as measure of central location or measure of center.

Mathematically mean can be defined as the sum of the all values in a given dataset divided by the number of observations in that data set under consideration. The mean is also called arithmetic mean or simply average.

Example: Consider the following data set consists of marks of 15 student in certain examination.

50, 55, 65, 43, 78, 20, 100, 5, 90, 23, 40, 56, 70, 88, 30

The mean of above data values is computed by adding all these values (50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 = 813) and then dividing by the number of observations added (15) which equals 54.2 marks, that is

$\frac{50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 }{15}=\frac{813}{15}=54.2$

The above procedure of calculating the mean can be represented mathematically

$\mu= \frac{\sum_{i=1}^n X_i}{N}$

The Greek symbol $\mu$ (pronounced “mu”) is the representation of population mean in statistics and $N$ is the number of observations in the population data set.

The above formula is known as population mean as it is computed for whole population. The sample mean can also be computed in same manner as population mean is computed. Only the difference is in representation of the formula, that is,

$\overline{X}= \frac{\sum_{i=1}^n X_i}{n}$.

The $\overline{X}$ is representation of sample mean and $n$ shows number of observations in the sample.

The mean is used for numeric data only. Statistically the data type for calculating mean should be Quantitative (variables should be measured on either ratio or interval scale), therefore, the numbers in data set can be continuous and/ or discrete in nature.

Note that mean should not be computed for alphabetic or categorical data (data should not belong to nominal or ordinal scale). Mean is influenced by very extreme values in data, i.e. very large or very small values in data changes the mean drastically.

For other measures of central tendencies visit: Measures of Central Tencencies