Introduction to Statistics

Statistics is the field focused on gathering, analyzing, interpreting, presenting, and organizing data. Essentially, it’s a branch of applied mathematics aimed at summarizing information. A key aspect of statistics involves dealing with uncertainty and variation, which are crucial for understanding different phenomena across various fields. By using statistical analysis, we can measure and interpret these uncertainties.

Fundamentals Of Statistics:

Statistics is a mathematical science including methods of collecting, organizing, and analyzing data so that meaningful conclusions can be drawn from them. In general, its investigations and analyses fall into two broad categories descriptive and inferential statistics.

1. Descriptive statistics deals with data processing without attempting to draw inferences. The data are presented in the form of tables and graphs. The characteristics of the data are described in simple terms. Events that are dealt with include everyday happenings such as accidents, prices of goods, business, incomes, epidemics, sports data, and population data.
2. Inferential statistics is a scientific discipline that uses mathematical tools to make forecasts and projections by analyzing the given data. This is of use to people employed in such fields as engineering, economics, biology, the social sciences, business, agriculture, and communications.

Data Collection Method:

1. Surveys: Surveys include collecting data from some individuals.

2. Experiments: Experiments include collecting the information but under some conditions.

3. Observational Studies: It includes collecting data without manipulating

Types Of Data:

1. Quantitative Data: It is the numeric data that is quantifiable.

2. Qualitative Data: It is non-numeric data that describes qualities or characteristics.

Fundamental of Descriptive Statistics:

An important characteristic of any set of data is the variation in the data. The degree to which the numerical data tends to vary about an average value is called the dispersion or scatteredness of the data.

The following are the measures of dispersion:

1. Range

2. Mean Deviation

3. Standard deviation and Variance

Range: The difference between maximum and minimum values

Variance: The mean of the squares of the deviations from the mean is called the variance and is denoted by σ² (read as sigma square). Variance is a quantity that leads to a proper measure of dispersion.

Standard Deviation: The standard deviation is a number that measures how far data values are from their mean.

Representations of Data:

Data can be represented using different graphs and charts like histograms, bar charts, pie charts, treemaps, etc.

1. Hstogram is used to display the frequency distribution of any value.

2. A scatter plot is used to show the relationship among two quantitative variables.

Solved Examples

Example 1: What is the range of the data $3,8,6,5,2,1,9,3,2$ ?
1) 9
2) 10
3) 8
4) 5

Solution:
Range :
The range is the difference between the smallest and largest observations. It is the simplest measure of Dispersion.

$
\text { Range }=9-1=8
$
Hence, the answer is an option 3.

Example 2: If the standard deviation of the numbers 2,3, a and 11 is 3.5, then which of the following is true?
1)$
3 a^2-26 a+55=0
$

2)$
3 a^2-32 a+84=0
$

3)$
3 a^2-34 a+91=0
$

4)$
3 a^2-23 a+44=0
$

Solution:
$
\begin{aligned}
& S D=\sqrt{\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2} \\
& \Rightarrow 3.5^2=\frac{49}{4}=\frac{4+9+a^2+121}{4}-\left(\frac{16+a}{4}\right)^2 \\
& \Rightarrow 3 a^2-32 a+84=0
\end{aligned}
$

Hence, the answer is the option 2.

Example 3: If the mean of the data : $7,8,9,7,8,7, \lambda, 8$ is 8 , then the variance of this data is :
1) $\frac{7}{8}$
2) 1
3) $\frac{9}{8}$
4) 2

Solution:
$
\begin{aligned}
& \text { mean of data }=\frac{7+8+9+7+8+7+7+8}{8}=8 \\
& \Rightarrow \lambda=10
\end{aligned}
$

Variance
$
\begin{aligned}
& V^2=\frac{(7-8)^2+(8-8)^2+(9-8)^2+(7-8)^2+0^2+(7-8)^2+(10-8)^2+(8-8)^2}{8} \\
& =\frac{8}{8}=1
\end{aligned}
$

Variance $=1$
Hence, the answer is the option 2.

Example 4: The mean of 5 observations is 5 and their variance is 124 . If three of the observations are 1,2 and 6 ; then the mean deviation from the mean of the data is :
1) 2.4
2) 2.8
3) 2.5
4) 2.6

Soluiton:
$
\begin{aligned}
& \frac{\sum x_i}{5}=5 \Rightarrow \sum x_i=25 \\
& \frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2=124 \\
& \frac{\sum x_i^2}{5}-25=124 \\
& \sum x_i^2=149 \times 5=745
\end{aligned}
$

Let the two observations be a \& b
$a+b+1+2+6=25$
$a+b=16$
$a^2+b^2+1^2+2^2+6^2=745$
$a^2+b^2+1+4+36=745$
$a^2+b^2=704$
Mean deviation $=\frac{\sum\left|x_i-5\right|}{5}=\frac{\left|x_1-5\right|+\left|x_2-5\right|+8}{5}$

$
=\frac{8+\left|x_1-5\right|+\left|11-x_1\right|}{5}=\frac{8+6}{5}=2.8
$

Hence, the answer is option (2).

Example 5: All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statisticis correct?
1) variance
2) mean
3) median
4) mode

Solution:
Mean, Mode, and Median are the measures of central tendency. All of these change with change in any observation.

Variance is the measure of the scattering of data. It is a measure of dispersion which do not change if every given observation changes by the same amount.

The measures of central tendency will change, but not measures of dispersion.
So variance will not change.

Hence, the answer is the option (1).

Summary

Statistics and probability are an important part of mathematics. These methods are widely used in real-life applications providing insights and solutions to complex problems. Mastery of these concepts can help in solving gaining deeper insights and contributing meaningfully to real-life problems.