Statistics

Statistics is a branch of mathematics, particularly applied mathematics. It is mainly used in differential and integral calculus, linear algebra, and probability theory. In statistics for data collection and analysis, they use different techniques like simple random, systematic, stratified, or cluster sampling.

Descriptive and inferential are the two types of statistics. Descriptive statistics summarizes and describes the characteristics of the data. It includes measures of central tendency and measures of dispersion. For instance, to analyze the performance of a class, we use descriptive statistics. In this case, it helps summarize the data using the average, variability, and test scores of students in the class. 

Inferential statistics is used to predict or generalize a large population with only a sample of data by using statistical techniques like hypothesis testing, analysis of variance, and regression analysis. For instance, inferential statistics helps us estimate the preference of students in a school to find their favorite subjects.  

Statistics is a branch of mathematics. Naturally, there will be similarities between math and statistics. Now, let’s learn a few differences between them. 
 

The fundamentals of statistics involve two key aspects. They are;
 

• Measures of central tendency
   
• Measures of dispersion 
   

The measures of central tendency include the mean, median, and mode, while the measures of dispersion consist of variance and standard deviation. 

The mean represents the average of all observations. The median is the middle value when observations are arranged in order from smallest to largest. The mode identifies the observation(s) that occur most frequently in a data set.

Variation measures how spread out a data set is. Standard deviation quantifies the dispersion of data points around the mean. The variance is equal to the square of the standard deviation, representing the average of the squared deviations from the mean.

Statistics is the study of data. There are two different types of statistics. Here, describing data characteristics falls under descriptive statistics, whereas inferential statistics involves drawing conclusions and making predictions from the data. Descriptive statistics summarize and organize data, while inferential statistics extend beyond the data to make generalizations about a larger population.

Descriptive statistics

Descriptive statistics use numerical calculations, graphs, or tables to summarize and represent data about a population. They provide a clear, concise graphical overview of the data, helping visualize and understand its main features and distribution.​

Descriptive statistics is primarily used to summarize objects or data. It is generally divided into two main categories.
 

• Measure of central tendency
   
• Measure of variability

Measure of central tendency: A measure of central tendency, also known as a summary statistic, is used to represent the central point or a typical value of a data set or sample. In statistics, the three commonly used measures of central tendency are: mean, median, and mode.
 

• Mean: Mean is the measure of the average value of all data points in a sample. It is calculated by dividing the sum of all values by the total number of values in the data set. The formula is expressed as:
  
  \(\text{Mean} = \frac{∑\text{values}}{\text{number of values}}\)
  
  This formula allows you to determine the central value that represents the data set as a whole.
   
• Median: The median is the central value in a data set when the numbers are arranged in either ascending or descending order. For datasets with an odd number of values, the median is the value exactly in the middle. When there are an even number of values, the median is the average of the two middle values. The formula for finding the median depends on whether the number of data points is odd or even.
  
  For an even number of values n, the median is calculated as the average of the \(\frac{n}{2}\)th term and the \((\frac{n}{2} +1)\)th term. For an odd number of values n, the median is the value at the \((\frac{n+1}{2})\) th position in the ordered data set.
   
• Mode: The mode is the value that occurs most frequently within a data set. A data set can have no mode (unimodal), a single mode (bimodal), or more than two modes (multimodal), depending on the frequency of values in the data.
   

Measure of variability: Measures of variability, also known as measures of dispersion, quantify how spread out data points are within a sample or population. The three common measures of variability in statistics are range, variance, and standard deviation. These measures describe the extent to which data values differ from the central value or mean, providing insight into the distribution's spread. 
 

• Range of data: The range is a measure that describes how spread out the values are within a data set or sample. It is calculated by subtracting the smallest value from the most significant value in the data set:
  
  \(\text{Range = Maximum value − Minimum value}\)
  
  This formula provides a simple way to understand the extent of variability or dispersion in the data.
   
• Variance: In probability theory and statistics, it measures the spread or dispersion of a data set. It is calculated as the average of the squared differences between each data point and the mean. Represented by the symbol σ2, variance quantifies how much the data points differ from the average value. A higher variance signifies that the data is more spread out, while a lower variance indicates that the data points are closer to the mean.
   
• Standard deviation: Standard deviation measures how spread out the values in a data set are around the mean. It calculates the average distance of each data point from the mean. The formula for standard deviation is:
  
  \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \)
  
  Where, s is the sample standard deviation, \(x_i\) is the i-th observation, \(\bar{x}\) is the sample mean, n is the number of observations.
  
  This measure helps quantify the variability within the data.
   
• Interquartile range (IQR): The interquartile range (IQR) is a measure of statistical dispersion that reflects the spread of the middle 50% of a data set. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1), representing the range within which the central half of the data lies. The IQR is useful for understanding data variability while reducing the impact of outliers and extreme values.
  
  Formula for IQR:
  
  \(IQR = Q_3 − Q_1\)
  
  Where:
  Q1 is the first quartile (25th percentile) and the median of the lower half of the data.
  Q3 is the third quartile (75th percentile) and the median of the upper half of the data.

Inferential statistics

Inferential statistics allows us to make predictions and draw conclusions about a population based on data collected from a sample. It generalizes findings from a smaller dataset and applies probabilities to estimate population parameters. Primarily, it is used to analyze and interpret results, perform hypothesis testing, and decide whether to reject the null hypothesis. This branch of statistics bridges descriptive statistics and broader data interpretation for informed decision-making.

Common types of inferential statistics, which are widely used and easy to interpret, include:
 

• One-sample hypothesis tests
• Confidence intervals
• Contingency tables and Chi-square tests
• T-tests and ANOVA
• Pearson correlation coefficient
• Bivariate regression
• Multivariate regression
   

These methods help analyze sample data and draw conclusions about larger populations.

Let us first understand what data representation is. It is a technical process of representing data visually. Now, before learning how we can represent data, we should be aware of the types of data: quantitative and qualitative. Qualitative data is a categorical data that describes the characteristics or categories such as gender, colors, names, etc. Quantitative data is the numerical data that involves measurable quantities and numbers such as weight, height, age, temperature, and so on. There are different types of data representation, such as:

• Bar graph
   
• Pie chart
   
• Line graph
   
• Pictograph
   
• Histogram
   
• Frequency distribution
   

Bar graph: The visual representation of data using rectangular bars. A bar graph can be either vertical or horizontal. 
 

Pie chart: It is a circular graph that is divided into sectors based on the data, each sector represents different data or categories. 
 

Line graph: A line graph is used to represent the data in series, which changes over time, and is connected with a straight line. 
 

Pictograph: In pictogram, data is represented using symbols, ideas, pictures, or objects. 
 

Histogram: A histogram may look like a bar graph, but it shows the frequency of continuous data. Unlike a bar graph, a histogram has bars that touch each other, indicating that the data has no gaps between intervals.
 

Frequency distribution: The table organizes data in ascending order, while the frequencies form a frequency distribution, showing how often the values appear. 

Data plays a major role in statistical analysis, so there are different methods to collect data in statistics. Let’s see some sampling techniques in statistics. 

Data plays a major role when it comes to statistical analysis, so there are different methods to collect data in statistics. Let’s see some sampling techniques here. 

Simple random sampling: In simple random sampling, the opportunity is given to the entire population to be selected for analysis. That is, based on chance, a few from the entire group are selected for the analysis. Let's say a class has 50 students, here, any five students can be chosen randomly. 

Systematic sampling: In systematic sampling, individuals are called at regular intervals from the starting point. For instance, in a line of 50 students, the individuals are decided randomly from a point. Example: From 100 students, select every 5th student to get 20 students.

Stratified sampling: In stratified sampling, a population is divided into subgroups based on shared characteristics. That is, 50 students are grouped based on gender, height, and weight. For example, in a school with 600 students (400 boys and 200 girls), select 60 students where 40 are boys and 20 are girls.

Cluster sampling: In cluster sampling, the population is divided into groups known as clusters, and some clusters are selected randomly for analysis. Example: A city has 20 schools. Randomly select 4 schools and survey all students in them.

The central tendency is a statistical measure that identifies a central point or typical value within a dataset. Measuring the central tendency in statistics is a part of descriptive statistics. It is finding the central value or the most repeating value in the given data set. The different measures of central tendency are:

• Arithmetic Mean

• Median

• Mode

• Geometric Mean

• Harmonic mean

Example: Find the mean, median, and mode of the dataset: 2, 4, 6, 8, 10

Let's first find the mean.

Mean: The average of all values.

 \(\text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6\).6

Median: The middle value when the data is arranged in the order.

Ordered data: 2, 4, 6, 8, 10 

Middle value = 6

Mode: It is the value that occurs most frequently. Since all values occur only once, there is no mode (or we say the data is mode-less).

The measure of dispersion is the way of spreading the data around the central value. It includes range, quartile deviation, mean deviation, and standard deviation. Now let’s see what these are in detail:

Range: The difference between the highest and the lowest values in a dataset.

Range = Max − Min

Quartile deviation: It measures the spread of the middle 50% of a data set. Here, we divide the data point into 4 quarters and find the median of the data points. The lower quartile (Q1) is the median of the lower half of the dataset, and the upper quartile (Q3) is the median of the upper half. The interquartile range is the difference between the upper and the lower quartile. 

Mean deviation: It is used to find how far each number is from the average and the average of the difference.

    \(\text{Mean Deviation} = \frac{\sum_{i=1}^{N} \lvert x_i - \mu \rvert}{N}\)

Standard deviation: It shows the deviation of numbers from the average.

Statistics is one of the fun and yet confusing concepts in mathematics. Here are some of the tips and tricks to master the concepts of statistics.
 

• Teachers should start teaching statistics with a few real-life examples to help young learners grasp the concepts. Using sports scores, we can ask them what the average number of goals our school team scores is.
   
• Children can learn concepts more quickly when they can see rather than read. Teachers and parents can use bar graphs, pie charts, line graphs, and histograms to teach statistics.
   
• Teachers can assign students a simple data-collection task and turn it into a game. Ask them to count everyone's shoe sizes in the classroom and let them choose how to sort and display the data.
   
• Start learning statistics with the easier concepts, then gradually increase the difficulty. Start by sorting and grouping, making tallies, reading simple graphs, mean, median, mode, probability basics, and advanced topics like variability and sampling. 
   
• Parents should encourage their children to predict answers. Before showing them the results, ask them questions like which color they think is the most occurring one

As we discussed, statistics is used in different fields, so let’s now see how statistics are used in our real life.  

• In health care, statistics are used for analyzing the effectiveness of treatments, designing clinical trials for new drugs or treatments, and so on. 

• The government uses statistics to plan and analyze public policy, population, resource allocation, and so on.

• In business, statistics are used to analyze and compare sales, market research, and finance. 

• Statistics is used to visualize the data using bar graphs, pie charts, and histogram for comparing and analyzing.