Definition of Statistics
Any raw Data, when collected and organized in the form of numerical or tables, is known as Statistics. Statistics is also the mathematical study of the probability of events occurring based on known quantitative Data or a Collection of Data.
Statistics attempts to infer the properties of a large Collection of Data from inspection of a sample of the Collection thereby allowing educated guesses to be made with a minimum of expense. There are generally 3 kinds of averages commonly used in Statistics. They are: (i) Mean, (ii) Median, and (iii) Mode.
Statistics is the study of Data Collection, Analysis, Interpretation, Presentation, and organizing in a specific way. Mathematical methods used for different analytics include mathematical Analysis, linear algebra, stochastic Analysis, the theory of measure-theoretical probability, and differential equations. Collecting, classifying, organizing, and displaying numerical Data is associated with Statistics. This helps one to grasp different outcomes from it and foresee several possibilities of various events. Statistics discuss information, observations, and Data in the form of numerical Data. We are able to find different indicators of central tendencies and the divergence of various values from the center with the help of Statistics.
The ability to analyze and interpret statistical Data is a vital skill for researchers and professionals from a wide variety of disciplines. You may need to make decisions on the basis of statistical Data, interpret statistical Data in research papers, do your own research, and interpret the Data.
Types of Statistics
There are two kinds of Statistics, which are descriptive Statistics and inferential Statistics. In descriptive Statistics, the Data or Collection Data are described in a summarized way, whereas in inferential Statistics, we make use of it in order to explain the descriptive kind. Both of them are used on a large scale. Also, there is another kind of Statistics where descriptive transitions into inferential Statistics.
Statistics is mainly divided into the following two categories.
Descriptive Statistics
Inferential Statistics
Descriptive Statistics
In the descriptive Statistics, the Data is described in a summarized way. The summarization is done from the sample of the population using different parameters like Mean or standard deviation. Descriptive Statistics are a way of using charts, graphs, and summary measures to organize, represent, and explain a set of Data.
Data is typically arranged and displayed in tables or graphs summarizing details such as histograms, pie charts, bars or scatter plots.
Descriptive Statistics are just descriptive and thus do not require normalization beyond the Data collected.
Inferential Statistics
In the Inferential Statistics, we try to interpret the Meaning of descriptive Statistics. After the Data has been collected, analyzed, and summarised we use Inferential Statistics to describe the Meaning of the collected Data.
Inferential Statistics use the probability principle to assess whether trends contained in the research sample can be generalized to the larger population from which the sample originally comes.
Inferential Statistics are intended to test hypotheses and investigate relationships between variables and can be used to make population predictions.
Inferential Statistics are used to draw conclusions and inferences, i.e., to make valid generalizations from samples.
Example
In a class, the Data is the set of marks obtained by 50 students. Now when we take out the Data average, the result is the average of 50 students’ marks. If the average marks obtained by 50 students are 88 out of 100, on the basis of the outcome, we will draw a conclusion.
Mean, Median and Mode in Statistics
Mean: Mean is considered the arithmetic average of a Data set that is found by adding the numbers in a set and dividing by the number of observations in the Data set.
Median: The middle number in the Data set while listed in either ascending or descending order is the Median.
Mode: The number that occurs the most in a Data set and ranges between the highest and lowest value is the Mode.
For n number of observations, we have
Mean = \[\overline {x} = \frac{\sum x}{n} = \sum {x}{n} \]
Median = \[ \frac{\left[ \frac {n}{2} + 1 \right ]^{th} term}{2}\] if n is odd.
Median = \[\frac {\left[ \frac {n}{2} \right ]^{th} term + \left[ \frac {n}{2} + 1 \right ]^{th} term }{2}\] if n is even.
Mode = The value which occurs most frequently
Measures of Dispersion in Statistics
The measures of central tendency do not suffice to describe the complete information about a given Data. Therefore, the variability is described by a value called the measure of dispersion.
The different measures of dispersion include:
The range in Statistics is calculated as the difference between the maximum value and the minimum value of the Data points.
The quartile deviation that measures the absolute measure of dispersion. The Data points are divided into 3 quarters. Find the Median of the Data points. The Median of the Data points to the left of this Median is said to be the upper quartile and the Median of the Data points to the right of this Median is said to be the lower quartile. Upper quartile - lower quartile is the interquartile range. Half of this is the quartile deviation.
The Mean deviation is the statistical measure to determine the average of the absolute difference between the items in a distribution and the Mean or Median of that series.
The standard deviation is the measure of the amount of variation of a set of values.
Solved Example
1. What is the probability of getting two tails and one head, when 3 coins are tossed at a time?
(A) 15
(B) 3/8
(C) 14
(D) 17
Solution:
Step 1: Number of possible outcomes when one coin is tossed = 2. Outcomes are HHH and TTT.
Step 2: The possible outcomes, when 3 coins are tossed are {TTT, THT, TTH, THH, HHT, HTH, HTT, HHH}.
Step 3: Number of favorable outcomes = 3. Favorable outcomes are THT, TTH, HTT.
Step 4: Substitute.
Step 5: So, the probability of getting two tails and one head is \[\frac{3}{8}\].
Correct answer: (B) \[\frac{3}{8}\].
Stages of Statistics
Collection of Data:
This is the first step of statistical Analysis where we collect the Data using different methods depending upon the case.
Organizing the Collected Data:
In the next step, we organize the collected Data in a Meaningful manner. All the Data is made easier to understand.
Presentation of Data:
In the third step we simplify the Data. These Data are presented in the form of tables, graphs, and diagrams.
Analysis of the Data:
Analysis is required to get the right results. It is often carried out using measures of central tendencies, measures of dispersion, correlation, regression, and interpolation.
Interpretation of Data:
In this last stage, conclusions are enacted. Use of comparisons is made. On this basis, forecasting is made.
Uses of Statistics
Statistics helps to obtain appropriate quantitative Data.
Statistics helps to present complex Data for the simple and consistent Interpretation of the Data in a suitable tabular, diagrammatic, and graphic form.
Statistics help to explain the nature and pattern of variability through quantitative observations of a phenomenon.
Statistics help to depict the Data in tabular form, or in a graphical form in order to understand it properly.
Applications of Statistics
Statistics is used in Machine Learning and Data Mining.
Statistics is used in Mathematics.
Statistics is used in Economics.
FAQs on Types of Statistics
1. What is statistics and what is its primary purpose in mathematics?
Statistics is a branch of mathematics focused on the **collection, analysis, interpretation, presentation, and organization of data**. Its primary purpose is to help us make sense of large amounts of information, identify patterns, and draw meaningful conclusions or make informed decisions from seemingly complex data sets.
2. What are the two main types or branches of statistics?
The two main types of statistics are **descriptive statistics** and **inferential statistics**. Descriptive statistics focuses on organizing, summarizing, and presenting data in a meaningful way, like calculating averages or creating graphs. Inferential statistics, however, uses data from a sample to make predictions or generalizations about a larger population.
3. How are different types of data categorized in statistics?
In statistics, data is broadly categorized into two main types: **qualitative data** and **quantitative data**. Qualitative data describes qualities or characteristics (like colours or opinions), while quantitative data deals with numerical values. Quantitative data is further divided into **discrete data** (countable items, like the number of students) and **continuous data** (measurable values within a range, like height or temperature).
4. What are the key methods or processes involved in statistical analysis?
The core methods involved in statistical analysis include:
- **Data Collection**: Gathering raw information from various sources.
- **Data Organization**: Arranging the collected data in a structured way, often in tables.
- **Data Presentation**: Displaying data using charts, graphs, or visual tools to make it understandable.
- **Data Analysis**: Applying mathematical techniques to interpret patterns and relationships within the data.
- **Data Interpretation**: Drawing conclusions and making inferences based on the analysis to solve problems or make predictions.
5. Can you provide some real-life examples of how statistics is used?
Statistics is applied in many everyday situations. For instance, calculating the **average marks** of students in a class, conducting surveys to **predict election results**, analyzing **weather patterns** to forecast future conditions, or determining the **effectiveness of a new medicine** through clinical trials are all practical applications of statistics.
6. Why is it important to identify the type of data before applying statistical methods?
It is crucial to identify the type of data (qualitative, quantitative, discrete, or continuous) before performing any statistical analysis because different data types require different **statistical tools and techniques**. Using the correct method ensures the analysis is accurate and that the conclusions drawn are valid and reliable.
7. How do descriptive and inferential statistics differ in their objectives?
**Descriptive statistics** aims to simply describe and summarize the main features of a dataset without making conclusions beyond the data itself. Its objective is to provide a clear picture of the information at hand. In contrast, **inferential statistics** aims to make predictions, generalizations, or draw conclusions about a larger population based on a smaller sample, often trying to infer what might be true beyond the observed data.
8. What is the role of probability in the field of statistics?
Probability plays a fundamental role in statistics, especially in **inferential statistics**. It helps in understanding the likelihood of events occurring and is essential for making predictions and assessing the confidence in statistical conclusions. When we draw conclusions about a larger group based on a small sample, probability helps us determine how likely our findings are to be accurate and not just due to chance.
9. How can statistics help in making informed decisions and predictions?
Statistics helps in making informed decisions and predictions by providing methods to **collect, analyze, and interpret data**. By understanding past trends and patterns within data, we can develop models to forecast future outcomes, evaluate risks, and quantify uncertainties. This allows individuals, businesses, and governments to make choices based on evidence rather than mere assumptions.
