Key Concepts in Statistics: From Distributions to Data Analysis
Binomial Distribution
Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution. This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure), given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial. The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive or independent of one another.
Sampling and Non-sampling Error
Sampling error refers to the difference between the sample statistics (such as the mean, proportion, or standard deviation) and the population parameters that they estimate. It occurs because a sample is a small subset of the population, and there is always some degree of variability between the sample statistics and the population parameters. The larger the sample size, the smaller the sampling error is likely to be.
Non-sampling Error
Non-sampling error refers to errors that occur during the data collection process, but are not due to the random selection of the sample. An example of non-sampling error would be if a survey is conducted to measure public opinion on a political issue, but the questions are worded in a biased or unclear way. This would lead to inaccurate or unreliable results, even if the sample was selected randomly.
Box and Whisker Plot
A Box and Whisker Plot, also called a Box Plot, is defined as a visual representation of the five-point summary. It consists of a rectangular “box” and two “whiskers.” A Box and Whisker Plot contains the following parts:
- Box: The box in the plot spans from the first quartile (Q1) to the third quartile (Q3). This box contains the middle 50% of the data and represents the interquartile range (IQR). The width of the box provides insights into the data’s spread.
- Whiskers: The whiskers extend from the minimum value to Q1 and from Q3 to the maximum value. They signify the range of the data, excluding potential outliers. The whiskers can vary in length, indicating the data’s skewness or symmetry.
Parameter and Statistic
Parameters:
- Parameters are fixed numerical values that describe a population.
- Parameters are estimated using statistical methods.
- Parameters are used to make inferences about the entire population.
Statistics:
- Statistics are calculated from a sample and describe a subset of the population.
- Statistics are directly computed from sample data.
- Statistics are used to make inferences about the sample and, by extension, the population from which it was drawn.
Census and Sample Survey
Census Survey
The process used in the census method includes the statistical compilation of all units or members of the target population under the survey. In this case, the population relates to the entire set of observations connected to a particular study. For instance, if students of a university have to give feedback on teaching faculty, the former will be held as the population of that study.
Sample Survey
The sample method chooses the different sample entities from the targeted population. This method involves a statistical analysis of an already determined number of observations that is derived from a larger set of populations. Sample methodology can be used of different kinds; these can be – simple random sampling or systematic sampling, cluster sampling or stratified sampling, etc. among others.
Moments
Moments in statistics are quantitative measures that describe the specific characteristics of a probability distribution. They help to understand the data set’s shape, spread, and central tendency. This article will discuss what moments in statistics are and four key moments: Mean, Variance & Standard Deviation, Skewness, and Kurtosis. Moments are usually meant to mean the average of the deviations of the observed values from any one particular value. Moments about the mean imply the average of deviations of the observed values from their mean and are of special interest in statistics. Now, if we dig deeper into it, there are various orders of moments. Order here refers to the ‘power’ of the difference of the data value from a particular value. Thus, the first-order moment about the mean is always zero.
Choice of Appropriate Measure of Central Tendency
A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.
The mean, median, and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode, and median, and learn how to calculate them and under what conditions they are most appropriate to be used.
Distinction Between Descriptive and Inferential Statistics
Descriptive Statistics
- It gives information about raw data, which describes the data in some manner.
- It helps in organizing, analyzing, and presenting data in a meaningful manner.
- It is used to describe a situation.
- It explains already known data and is limited to a sample or population having a small size.
- It can be achieved with the help of charts, graphs, tables, etc.
Inferential Statistics
- It makes inferences about the population using data drawn from the population.
- It allows us to compare data and make hypotheses and predictions.
- It is used to explain the chance of occurrence of an event.
- It attempts to reach a conclusion about the population.
- It can be achieved by probability.
Regression
Some of the key points of Regression are as given below:
- Regression is a statistical technique that relates a dependent variable to one or more independent variables.
- A regression model can show whether changes observed in the dependent variable are associated with changes in one or more of the independent variables.
- It does this by essentially determining a best-fit line and seeing how the data is dispersed around this line.
- Regression helps economists and financial analysts in things ranging from asset valuation to making predictions.
- For regression results to be properly interpreted, several assumptions about the data and the model itself must hold.
Role of Statistics in Information Technology
Statistics plays a crucial role in information technology (IT) by enabling data-driven decision-making and optimizing processes. Here are the key points:
- Data Analysis and Interpretation: Statistics helps in analyzing large datasets to extract meaningful insights, identify trends, and predict outcomes, which is essential in IT for business intelligence and analytics.
- Algorithm Design: Statistical methods are fundamental in developing algorithms for machine learning, artificial intelligence, and data mining, which rely heavily on probabilistic and statistical models.
- Performance Optimization: IT systems’ performance, such as network efficiency and software reliability, can be evaluated and improved using statistical tools.
- Quality Assurance: Statistics is used in IT to test software and hardware, ensuring quality through defect detection and reliability assessments.
- Risk Management: Statistical models help predict and mitigate risks, such as cybersecurity threats or system failures, ensuring robust IT infrastructure.
Properties of Correlation
Here are the key properties of correlation:
- Range of Values: The correlation coefficient r lies between -1 and +1.
- r = +1: Perfect positive correlation.
- r = -1: Perfect negative correlation.
- r = 0: No correlation.
- Symmetry: The correlation between two variables X and Y is the same as between Y and X, i.e., r(X,Y) = r(Y,X).
- Unitless Measure: The correlation coefficient is a dimensionless number, meaning it does not depend on the units of measurement of the variables.
- Linear Relationship: Correlation measures the strength and direction of a linear relationship between two variables. It may not capture non-linear relationships effectively.
- Independence vs. Correlation: A correlation of r = 0 indicates no linear relationship, but it does not necessarily mean the variables are independent; they could have a non-linear relationship.
Random Variables
A random variable is a fundamental concept in statistics that bridges the gap between theoretical probability and real-world data. A random variable in statistics is a function that assigns a real value to an outcome in the sample space of a random experiment. For example, if you roll a die, you can assign a number to each possible outcome.
There are two basic types of random variables:
- Discrete Random Variables (which take on specific values)
- Continuous Random Variables (assume any value within a given range)
Binomial Distribution
Binomial distribution is a fundamental probability distribution in statistics, used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is particularly useful when you want to calculate the probability of a specific number of successes, such as flipping coins, quality control in manufacturing, or predicting survey outcomes.
Measures of Dispersion and Their Suitability
Dispersion in statistics is a way to describe how spread out or scattered the data is around an average value. It helps to understand if the data points are close together or far apart.
Dispersion shows the variability or consistency in a set of data. There are different measures of dispersion like range, variance, and standard deviation.
Types of Measures of Dispersion
Measures of dispersion can be classified into the following two types:
- Absolute Measure of Dispersion
- Relative Measure of Dispersion
Some absolute measures of dispersion are:
- Range: It is defined as the difference between the largest and the smallest value in the distribution.
- Mean Deviation: It is the arithmetic mean of the difference between the values and their mean.
- Standard Deviation: It is the square root of the arithmetic average of the square of the deviations measured from the mean.
- Variance: It is defined as the average of the square deviation from the mean of the given data set.
- Quartile Deviation: It is defined as half of the difference between the third quartile and the first quartile in a given data set.
- Interquartile Range: The difference between the upper (Q3) and lower (Q1) quartile is called the Interquartile Range. Its formula is given as Q3 – Q1.
What is Data?
Data can be defined as a systematic record of a particular quantity. It is the different values of that quantity represented together in a set. It is a collection of facts and figures to be used for a specific purpose such as a survey or analysis. When arranged in an organized form, it can be called information. The source of data (primary data, secondary data) is also an important factor.
Estimation
Estimation in statistics are any procedures used to calculate the value of a population drawn from observations within a sample size drawn from that population. There are two types of estimation: either point or interval estimation.
Difference Between Primary Data and Secondary Data
Primary Data
- Primary data is the first data collected by a researcher for the first time.
- Primary data is called real-time data.
- The process is very much involved in collecting primary data.
- Primary data is expensive.
- The primary data takes a long time for collection.
- Primary data is available in crude form.
- Primary data is more accurate than secondary data.
- Primary data is more reliable than secondary data.
- There is also difficulty in collecting data.
- The data is always specific to the researcher’s need.
Secondary Data
- Whereas secondary data is data that has already been collected by someone earlier.
- While this is not real-time data, it is related to the past.
- While collecting secondary data, it does not involve much process but rather quickly and easily.
- While it is economical.
- While secondary data takes a shorter time than primary data for collection.
- While it is available in a processed or refined form.
- While it is less accurate than primary data.
- While secondary data is less reliable than primary data.
- While there is no difficulty in collecting data.
- The data collected may or may not be specific to the researcher’s need.
Distinction Between Absolute and Relative Measures of Dispersion
Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The normal distribution appears as a”bell curv” when graphed.
Some of the key points of Normal Distribution are as given below:
- The normal distribution is the proper term for a probability bell curve.
- In a normal distribution, the mean is zero and the standard deviation is 1. It has zero skew and a kurtosis of 3.
- Normal distributions are symmetrical, but not all symmetrical distributions are normal.