Descriptive Statistics

Descriptive statistics is a subfield of statistics that deals with characterizing the features of known data. Descriptive statistics give summaries of either population or sample data. Aside from descriptive statistics, inferential statistics is another important discipline of statistics used to draw conclusions about population data.

Descriptive statistics is divided into two categories:

• Measures of Central Tendency
• Measures of Dispersion

In this article, we will learn about descriptive statistics, including their many categories, formulae, and examples in detail.

What is Descriptive Statistics?

Descriptive statistics is a branch of statistics focused on summarizing, organizing, and presenting data in a clear and understandable way. Its primary aim is to define and analyze the fundamental characteristics of a dataset without making sweeping generalizations or assumptions about the entire data set.

The main purpose of descriptive statistics is to provide a straightforward and concise overview of the data, enabling researchers or analysts to gain insights and understand patterns, trends, and distributions within the dataset.

Descriptive statistics typically involve measures of central tendency (such as mean, median, mode), dispersion (such as range, variance, standard deviation), and distribution shape (including skewness and kurtosis). Additionally, graphical representations like charts, graphs, and tables are commonly used to visualize and interpret the data.

Histograms, bar charts, pie charts, scatter plots, and box plots are some examples of widely used graphical techniques in descriptive statistics.

Descriptive Statistics Definition

Descriptive statistics is a type of statistical analysis that uses quantitative methods to summarize the features of a population sample. It is useful to present easy and exact summaries of the sample and observations using metrics such as mean, median, variance, graphs, and charts.

Types of Descriptive Statistics

There are three types of descriptive statistics:

• Measures of Central Tendency
• Measures of Dispersion
• Measures of Frequency Distribution

Measures of Central Tendency

The central tendency is defined as a statistical measure that may be used to describe a complete distribution or dataset with a single value, known as a measure of central tendency. Any of the central tendency measures accurately describes the whole data distribution. In the following sections, we will look at the central tendency measures, their formulae, applications, and kinds in depth.

• Mean
• Median
• Mode

Mean

Mean is the sum of all the components in a group or collection divided by the number of items in that group or collection. Mean of a data collection is typically represented as x̄ (pronounced “x bar”). The formula for calculating the mean for ungrouped data to express it as the measure is given as follows:

For a series of observations:

x̄ = Σx / n

where,

• x̄ = Mean Value of Provided Dataset
• Σx = Sum of All Terms
• n = Number of Terms

Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 66 and 50. Determine the mean weight for the provided collection of data.

Mean = Σx/n

= (54 + 32 + 45 + 61 + 20 + 66 + 50)/7

= 328 / 7

= 46.85

Thus, the group’s mean weight is 46.85 kg.

Median

Median of a data set is the value of the middle-most observation obtained after organizing the data in ascending order, which is one of the measures of central tendency. Median formula may be used to compute the median for many types of data, such as grouped and ungrouped data.

Ungrouped Data Median (n is odd): [(n + 1)/2]^th term

Ungrouped Data Median (n is even): [(n / 2)^th term + ((n / 2) + 1)^th term]/2

where,

• n = Number of Terms

Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 66 and 50. Determine the median weight for the provided collection of data.

Arrange the provided data collection in ascending order: 20, 32, 45, 50, 54, 61, 66

Median = [(n + 1) / 2]^th term

= [(7 + 1) / 2]^th term

= 4^th term

= 50

Thus, group’s median weight is 50 kg.

Mode

Mode is one of the measures of central tendency, defined as the value that appears the most frequently in the provided data, i.e. the observation with the highest frequency is known as the mode of data. The mode formulae provided below can be used to compute the mode for ungrouped data.

Mode of Ungrouped Data: Most Repeated Observation in Dataset

Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 45 and 50. Determine the mode weight for the provided collection of data.

Mode = Most repeated observation in Dataset

= 45

Thus, group’s mode weight is 45 kg.

Measures of Dispersion

If the variability of data within an experiment must be established, absolute measures of variability should be employed. These metrics often reflect differences in a data collection in terms of the average deviations of the observations. The most prevalent absolute measurements of deviation are mentioned below. In the following sections, we will look at the variability measures, their formulae in depth.

• Range
• Standard Deviation
• Variance

Range

The range represents the spread of your data from the lowest to the highest value in the distribution. It is the most straightforward measure of variability to compute. To get the range, subtract the data set’s lowest and highest values.

Range = Highest Value – Lowest Value

Example: Calculate the range of the following data series:  5, 13, 32, 42, 15, 84

Arrange the provided data series in ascending order: 5, 13, 15, 32, 42, 84

Range = H – L

= 84 – 5

= 79

So, the range is 79.

Standard Deviation

Standard deviation (s or SD) represents the average level of variability in your dataset. It represents the average deviation of each score from the mean. The higher the standard deviation, the more varied the dataset is.

To calculate standard deviation, follow these six steps:

Step 1: Make a list of each score and calculate the mean.

Step 2: Calculate deviation from the mean, by subtracting the mean from each score.

Step 3: Square each of these differences.

Step 4: Sum up all squared variances.

Step 5: Divide the total of squared variances by N-1.

Step 6: Find the square root of the number that you discovered.

Example: Calculate standard deviation of the following data series:  5, 13, 32, 42, 15, 84.

Solution:

Step 1: First we have to calculate the mean of following series using formula: Σx / n

Step 2: Now calculate the deviation from mean, subtract the mean from each series.

Step 3: Squared the deviation from mean and then add all the deviation.

Series	Deviation from Mean	Squared Deviation
5	5-31.83 = -26.83	719.85
13	13-31.83 = -18.83	354.57
32	32-31.83 = 0.17	0.0289
42	42-31.83 = 10.17	103.43
15	15-31.83 = -16.83	283.25
84	84-31.83 = 52.17	2721.71
Mean = 191/6 = 31.83	sum = 0	Sum = 4182.84

Step 4: Divide the squared deviation with N-1 => 4182.84 / 5 = 836.57

Step 5: √836.57 = 28.92

So, the standard deviation is 28.92

Variance

Variance is calculated as average of squared departures from the mean. Variance measures the degree of dispersion in a data collection. The more scattered the data, the larger the variance in relation to the mean. To calculate the variance, square the standard deviation.

Symbol for variance is s²

Example: Calculate the variance of the following data series:  5, 13, 32, 42, 15, 84.

Solution:

First we have to calculate the standard deviation, that we calculate above i.e.

SD = 28.92

s² = (SD)² = (28.92)² = 836.37

So, the variance is 836.37

Mean Deviation

Mean Deviation is used to find the average of the absolute value of the data about the mean, median, or mode. Mean Deviation is some times also known as absolute deviation. The formula mean deviation is given as follows:

Mean Deviation = ∑ⁿ₁ |X – μ|/n

where,

• μ is Central Value

Quartile Deviation

Quartile Deviation is the Half of difference between the third and first quartile. The formula for quartile deviation is given as follows:

Quartile Deviation = (Q₃ − Q₁)/2

where,

• Q₃ is Third Quartile
• Q₁ is First Quartile

Other measures of dispersion include the relative measures also known as the coefficients of dispersion.

Measures of Frequency Distribution

Datasets consist of various scores or values. Statisticians employ graphs and tables to summarize the occurrence of each possible value of a variable, often presented in percentages or numerical figures.

For instance, suppose you were conducting a poll to determine people’s favorite Beatles. You would create one column listing all potential options (John, Paul, George, and Ringo) and another column indicating the number of votes each received. Statisticians represent these frequency distributions through graphs or tables

Univariate Descriptive Statistics

Univariate descriptive statistics focus on one thing at a time. We look at each thing individually and use different ways to understand it better. Programs like SPSS and Excel can help us with this.

If we only look at the average (mean) of something, like how much people earn, it might not give us the true picture, especially if some people earn a lot more or less than others. Instead, we can also look at other things like the middle value (median) or the one that appears most often (mode). And to understand how spread out the values are, we use things like standard deviation and variance along with the range.

Bivariate Descriptive Statistics

When we have information about more than one thing, we can use bivariate or multivariate descriptive statistics to see if they are related. Bivariate analysis compares two things to see if they change together. Before doing any more complicated tests, it’s important to look at how the two things compare in the middle.

Multivariate analysis is similar to bivariate analysis, but it looks at more than two things at once, which helps us understand relationships even better.

Representations of Data in Descriptive Statistics

Descriptive statistics use a variety of ways to summarize and present data in an understandable manner. This helps us grasp the data set’s patterns, trends, and properties.

Frequency Distribution Tables: Frequency distribution tables divide data into categories or intervals and display the number of observations (frequency) that fall into each one. For example, suppose we have a class of 20 students and are tracking their test scores. We may make a frequency distribution table that contains score ranges (e.g., 0-10, 11-20) and displays how many students scored in each range.

Graphs and Charts: Graphs and charts graphically display data, making it simpler to understand and analyze. For example, using the same test score data, we may generate a bar graph with the x-axis representing score ranges and the y-axis representing the number of students. Each bar on the graph represents a score range, and its height shows the number of students scoring within that range.

These approaches help us summarize and visualize data, making it easier to discover trends, patterns, and outliers, which is critical for making informed decisions and reaching meaningful conclusions in a variety of sectors.

Descriptive Statistics Applications

Descriptive statistics are used in a variety of sectors to summarize, organize, and display data in a meaningful and intelligible way. Here are a few popular applications:

• Business and Economics: Descriptive statistics are useful for analyzing sales data, market trends, and customer behaviour. They are used to generate averages, medians, and standard deviations in order to better evaluate product performance, pricing strategies, and financial metrics.
• Healthcare: Descriptive statistics are used to analyze patient data such as demographics, medical histories, and treatment outcomes. They assist healthcare workers in determining illness prevalence, assessing treatment efficacy, and identifying risk factors.
• Education: Descriptive statistics are useful in education since they summarize student performance on tests and examinations. They assist instructors in assessing instructional techniques, identifying areas for improvement, and monitoring student growth over time.
• Market Research: Descriptive statistics are used to analyze customer preferences, product demand, and market trends. They enable businesses to make educated decisions about product development, advertising campaigns, and market segmentation.
• Finance and investment: Descriptive statistics are used to analyze stock market data, portfolio performance, and risk management. They assist investors in determining investment possibilities, tracking asset values, and evaluating financial instruments.

Difference Between Descriptive Statistics and Inferential Statistics

Difference between Descriptive Statistics and Inferential Statistics is studied using the table added below as,

Descriptive Statistics vs Inferential Statistics
Descriptive Statistics	Inferential Statistics
Does not need making predictions or generalizations outside the dataset.	This involves making forecasts or generalizations about a wider population.
Gives basic summary of the sample.	Concludes about the population based on the sample.
Examples include mean, median, mode, standard deviation, etc.	Examples include hypothesis testing, confidence intervals, regression analysis, etc.
Focuses on the properties of the current dataset.	Concentrates on drawing conclusions about the population from sample data.
Helpful for comprehending data patterns and linkages.	Useful for making judgements, predictions, and drawing inferences that go beyond the observed facts.

Example of Descriptive Statistics Examples

Example 1: Calculate the Mean, Median and Mode for the following series: {4, 8, 9, 10, 6, 12, 14, 4, 5, 3, 4}

Solution:

First, we are going to calculate the mean.

Mean = Σx / n

= (4 + 8 + 9 + 10 + 6 + 12 + 14 + 4 + 5 + 3 + 4)/11

= 79 / 11

= 7.1818

Thus, the Mean is 7.1818.

Now, we are going to calculate the median.

Arrange the provided data collection in ascending order: 3, 4, 4, 4, 5, 6, 8, 9, 10, 12, 14

Median = [(n + 1) / 2]^th term

= [(11 + 1) / 2]^th term

= 6^th term

= 6

Thus, the median is 6.

Now, we are going to calculate the mode.

Mode = The most repeated observation in the dataset

= 4

Thus, the mode is 4.

Example 2: Calculate the Range for the following series: {4, 8, 9, 10, 6, 12, 14, 4, 5, 3, 4}

Solution:

Arrange the provided data series in ascending order: 3, 4, 4, 4, 5, 6, 8, 9, 10, 12, 14

Range = H – L

= 14 – 3

= 11

So, the range is 11.

Example 3: Calculate the standard deviation and variance of following data: {12, 24, 36, 48, 10, 18}

Solution:

First we are going to compute standard deviation. For standard deviation calculate the mean, deviation from mean and squared deviation.

Series	Deviation from Mean	Squared Deviation
12	12-24.66 = -12.66	160.28
24	24-24.66 = -0.66	0.436
36	36-24.66 = 11.34	128.595
48	48-24.66 = 23.34	544.76
10	10-24.66 = -14.66	214.92
18	18-24.66 = -6.66	44.36
Mean = 148/6 = 24.66	sum = 0	Sum = 1093.351

Dividing squared deviation with N-1 => 1093.351 / 5 = 218.67

√(218.67) = 14.79

So, the standard deviation is 14.79.

Now we are going to calculate the variance.

s = 14.79

s² = 218.744

So, the variance is 218.744

Practice Problems on Descriptive Statistics

P1) Determine the sample variance of the following series: {17, 21, 52, 28, 26, 23}

P2) Determine the mean and mode of the following series: {21, 14, 56, 41, 18, 15, 18, 21, 15, 18}

P3) Find the median of the following series: {7, 24, 12, 8, 6, 23, 11}

P4) Find the standard deviation and variance of the following series: {17, 28, 42, 48, 36, 42, 20}

FAQs of Descriptive Statistics

What is meant by descriptive statistics?

Descriptive statistics seek to summarize, organize, and display data in an accessible manner while avoiding making sweeping generalizations about the whole population. It aids in discovering patterns, trends, and distributions within the collection.

How is the mean computed in descriptive statistics?

Mean is computed by adding together all of the values in the dataset and dividing them by the total number of observations. It measures the dataset’s central tendency or average value.

What role do measures of variability play in descriptive statistics?

Measures of variability, such as range, standard deviation, and variance, aid in quantifying the spread or dispersion of data points around the mean. They give insights on the dataset’s variety and consistency.

Can you explain the median in descriptive statistics?

The median is the midpoint value of a dataset whether sorted ascending or descending. It measures central tendency and is important when dealing with skewed data or outliers.

How can frequency distribution measurements contribute to descriptive statistics?

Measures of frequency distribution summarize the incidence of various values or categories within a dataset. They give insights into the distribution pattern of the data and are commonly represented by graphs or tables.

How are inferential statistics distinguished from descriptive statistics?

Inferential statistics use sample data to draw inferences or make predictions about a wider population, whereas descriptive statistics summarize aspects of known data. Descriptive statistics concentrate on the present dataset, whereas inferential statistics go beyond the observable data.

Why are descriptive statistics necessary in data analysis?

Descriptive statistics give researchers and analysts a clear and straightforward summary of the dataset, helping them to identify patterns, trends, and distributions. It aids in making educated judgements and gaining valuable insights from data.

What are the four types of descriptive statistics?

There are four major types of descriptive statistics:

• Measures of Frequency
• Measures of Central Tendency
• Measures of Dispersion or Variation
• Measures of Position

Which is an example of descriptive statistics?

Descriptive statistics examples include the study of mean, median, and mode.