Introduction to Statistics

Statistics is the field focused on collecting, 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.

This Story also Contains

1. Statistics
2. Data
3. Measures of Central Tendency
4. Measures of Dispersion
5. Representation of Data
6. Solved Examples on Statistics
7. Summary

This article is about the concept fundamentals of Statistics. This is an important concept which falls under the broader category of Statistics. This is not only important for board exams but also for various competitive exams.

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

Any bit of information is data. For example, the marks you obtained in your Math exam are data.

Data is a collection of information, measurements or observations. The data once collected must be arranged or organized in a way so that inferences or conclusions can be made out from it.

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.

Measures of Central Tendency

A measures of central tendency (or central value) is a single value that attempts to describe a set of data by identifying the central position within that set of data. Apart from mean (often called the average), there are other central values such as the median and the mode. The valid measures of central tendency are

• Mean
• Median
• Mode

Measures of Dispersion

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:

• Range

• Mean Deviation

• Standard deviation

• Variance

Representation of Data

The data once collected must be arranged or organized in a way so that inferences or conclusions can be made out from it. The frequency of any value is the number of times that value appears in a data set.

The following are the ways to for the representation of data,

1. Ungrouped distribution
2. Ungrouped frequency distribution
3. Grouped frequency distribution

Recommended Video Based on Statistics

Solved Examples on Statistics

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 $a$ and $b$
$a+b+1+2+6=25$
$a+b=16$
$a^2+b^2+1^2+2^2+6^2=745$
$a^2+b^2+1+4+36=745$
$a^2+b^2=704$
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 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.