Exploring the Basics of Descriptive Statistics: Part One | by Nayanshewkani | Mar, 2023

Statistics is the study of collecting, analyzing, and interpreting numerical data in order to make informed decisions or draw conclusions about a larger population or phenomenon. It involves using mathematical and computational tools to organize and summarize data, identify patterns and relationships, and make predictions or inferences based on the available evidence.

Statistics plays a crucial role in data analysis across various fields, such as:

1. Business: Statistics are used in market research to analyze customer behavior, preferences, and buying patterns to help companies make informed decisions about product development, pricing, and marketing strategies.
2. Healthcare: Medical researchers use statistics to analyze patient data and conduct clinical trials to identify effective treatments for various diseases.
3. Education: Statistics are used in educational research to analyze student performance and assess the effectiveness of teaching methods and educational programs.
4. Government: Statistics are used in public policy research to analyze the effectiveness of government programs and policies.
5. Social Sciences: Statistics are used in social science research to analyze data from surveys, experiments, and observational studies to understand human behavior, attitudes, and social trends.
6. Environmental Science: Statistics are used in environmental research to analyze data on climate, weather patterns, and pollution levels to identify trends and patterns.

Overall, statistics provide a powerful set of tools for data analysis and help researchers make sense of complex data sets and draw meaningful conclusions from their findings.

There are two main types of statistics: descriptive statistics and inferential statistics.

1. Descriptive statistics: Descriptive statistics are used to describe and summarize data in a meaningful way. They include measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and graphical representations of data (histograms, bar charts, scatterplots).
2. Inferential statistics: Inferential statistics are used to make predictions or draw conclusions about a larger population based on a sample of data. They involve hypothesis testing, confidence intervals, and regression analysis.

There are two main aspects in Statistical terms that have more prevalence in making predictions using Statistical Analysis:

1. Population refers to the entire group of individuals or objects that researchers want to study and draw conclusions about. For example, if a researcher is interested in studying the average height of all adults in a particular country, the population would be all adults living in that country.
2. Sample refers to a smaller subset of the population that researchers actually collect data from in order to draw conclusions about the larger population. For example, if a researcher wants to estimate the average height of all adults in a particular country, it may be impractical or impossible to measure the height of every single adult in the population. Instead, the researcher may randomly select a sample of, say, 1000 adults from different regions of the country and measure their heights. The sample would then be used to make inferences about the average height of the entire population of adults in that country.

When creating a sample from a population, there are several things to be careful about to ensure that the sample is representative of the population and that the conclusions drawn from the sample can be generalized to the larger population. Some of the things to be careful about are:

1. Sampling bias: Sampling bias occurs when the sample is not representative of the population. This can happen when certain individuals or groups are more likely to be included in the sample than others, leading to results that may not accurately reflect the entire population.
2. Sample size: Sample size is an important consideration when creating a sample. A larger sample size generally leads to more accurate estimates of population parameters. However, a larger sample size may not always be practical or feasible.
3. Sampling method: There are different methods for creating a sample, such as random sampling, stratified sampling, and cluster sampling. The sampling method used should be appropriate for the research question and the characteristics of the population.
4. Measurement error: Measurement error occurs when the data collected from the sample are not accurate or precise. This can lead to inaccurate estimates of population parameters and affect the validity of the conclusions drawn from the sample.

Types of data in Statistics:

1. Nominal data: Nominal data are categorical data that cannot be ranked or ordered.
2. Ordinal data: Ordinal data are categorical data that can be ranked or ordered.
3. Discrete data: Discrete data are countable and take on only certain values.
4. Continuous data: Continuous data can take on any value within a range and can be measured with a degree of precision.

A measure of central tendency is a statistical measure that represents the typical or central value of a dataset. It provides a summary of the distribution of data by describing where the center of the data lies. The three main measures of central tendency are the mean, median, and mode.

1. Mean: The mean is the sum of all the values in a dataset divided by the number of values. It is the most commonly used measure of central tendency and is often referred to as the “average”. However, the mean is affected by extreme values or outliers and may not be representative of the typical value of the data.
2. Median: The median is the middle value of a dataset when the values are arranged in order. It is not affected by extreme values or outliers and is often used when the distribution of data is skewed or contains outliers.
3. Mode: The mode is the most frequent value in a dataset. It is useful for describing the most common value in a dataset and can be used for both nominal and ordinal data.

In summary, measures of central tendency are used to provide a summary of the center or typical value of a dataset. The choice of measure depends on the nature of the data and the research question being addressed.

Weighted Mean

The weighted mean is a type of average that takes into account the importance or weight of each value in a dataset. The weighted mean is calculated by multiplying each value by its corresponding weight, summing these products, and then dividing by the sum of the weights.

For example, consider a dataset of exam scores for a class of 30 students, where each student’s score is weighted according to their performance in the class:

To calculate the weighted mean of the exam scores, we first multiply each score by its corresponding weight:

(80 x 0.2) + (75 x 0.3) + (90 x 0.5) = 16 + 22.5 + 45 = 83.5

Next, we divide this sum by the total weight:

0.2 + 0.3 + 0.5 = 1

Weighted mean = 83.5 / 1 = 83.5

In this example, the weighted mean of the exam scores is 83.5. This means that, on average, the class performed better than a student who scored exactly 83.5 on the exam. The weighted mean takes into account the fact that some students’ scores are more important or influential than others due to their relative weights.

A measure of dispersion is a statistical measure that describes the spread or variability of a dataset. It provides information about how tightly or loosely the values in a dataset are clustered around the central value (i.e., the measure of central tendency). The three main measures of dispersion are range, variance, and standard deviation.

Range

Range is a measure of dispersion in statistics that represents the difference between the largest and smallest values in a dataset. It provides a simple measure of the spread or variability of the data but is sensitive to outliers and extreme values.

To calculate the range of a dataset, you simply subtract the smallest value from the largest value:

Range = Largest value — smallest value

For example, consider the following dataset of test scores:

The largest value in this dataset is 90, and the smallest value is 65. Therefore, the range of the dataset is:

Range = 90–65 = 25

This means that the scores in this dataset range from 65 to 90, with a spread of 25 points.

The range can be a useful measure of dispersion for datasets with relatively few values, or when a quick estimate of the spread is needed. However, it is not a robust measure of dispersion since it is sensitive to outliers and extreme values, which can skew the range and make it an unreliable measure of the typical spread of the data.

Variance

Variance is a statistical measure that describes how spread out the values in a dataset are from the mean or central value. It is one of the most commonly used measures of dispersion in statistics and provides a way to quantify the amount of variability or diversity in the data.

The variance of a dataset is calculated as the average of the squared differences between each value in the dataset and the mean of the dataset. The formula for variance is as follows:

Population Variance = ∑(xi — μ)² / (N)

Sample Variance = ∑(xi — x)² / (n-1)

Where:

• xi is the ith value in the dataset
• μ is the mean of the dataset
• n is the number of values in the dataset
• ∑ is the sum of the values from i=1 to i=n

The variance measures the average distance of each data point from the mean of the dataset. The squared differences are used to eliminate the possibility of negative values in the calculation, and dividing by (n-1) instead of n is known as Bessel’s correction and is used to provide an unbiased estimate of the population variance.

Standard Deviation

Standard deviation is a widely used measure of the dispersion or variability of a dataset. It is calculated as the square root of the variance and provides a way to measure how far the data points are from the mean.

The formula for standard deviation is:

Standard deviation = square root of variance = √(∑(xi — μ)² / (n-1))

Where:

• xi is the ith value in the dataset
• μ is the mean of the dataset
• n is the number of values in the dataset
• ∑ is the sum of the values from i=1 to i=n

Like variance, standard deviation measures the average distance of each data point from the mean of the dataset. However, standard deviation is expressed in the same units as the original data, making it easier to interpret and compare across different datasets.

Coefficient of Variation

The coefficient of variation (CV) is a relative measure of the variability of a dataset, expressed as a percentage of the mean. It is often used to compare the variability of different datasets that may have different units or scales.

The formula for the coefficient of variation is:

CV = (standard deviation / mean) x 100%

Where:

• standard deviation is the standard deviation of the dataset
• mean is the mean of the dataset

Descriptive statistics is a fundamental tool in data analysis, providing a way to summarize and describe the main features of a dataset. In this part, we covered some of the basic concepts of descriptive statistics, including population and sample, types of data, measures of central tendency, and measures of dispersion. These concepts provide a foundation for further exploration of descriptive statistics and their application in various fields, from business to science and beyond. In the next part, we will dive deeper into some of the advanced topics.