Descriptive Statistics Categories

Descriptive statistics is a type of statistics that focuses on characterizing the properties of known data. Descriptive statistics give summaries of data from the population or a sample. Inferential statistics, in addition to descriptive statistics, is an important field of statistics that is used to make inferences about population data. Measures of central tendency and measures of dispersion are the two basic types of descriptive statistics.

Descriptive Statistics: 

Descriptive statistics are used to summarise the characteristics of a sample in a quantitative or visual way. Data from a sample may be studied using certain tools to detect certain trends or patterns. It aids in the organization of data in a manner that is more manageable and understandable.

Definition of Descriptive Statistics:

Descriptive statistics may be described as a branch of statistics that employs quantitative approaches to summarise the features of a sample. Using measurements like mean, median, variance, graphs, and charts, assists in providing clear and exact summaries of the sample and observations. Data with only one variable is described using univariate descriptive statistics. Bivariate and multivariate descriptive statistics, on the other hand, are used to describe data with several variables.

Types of Descriptive Statistics:

Measures of central tendency and measures of dispersion are two forms of descriptive statistics used to characterize the features of grouped and ungrouped data quantitatively. The raw data obtained from an experiment is referred to as ungrouped data. The term “grouped data” refers to data that has been logically ordered. Graphs, charts, and tables are used in descriptive statistics to graphically depict data.

Central Tendency Measures:

Measures of central tendency are used in descriptive statistics to characterize data by finding a single representative center value. The following are the most significant metrics of central tendency:

1. Mean: The mean is calculated by dividing the total number of observations by the sum of all observations. The mean is calculated using the following formulas:

Ungrouped data Mean: x̄ = Σxi / n

Grouped data Mean: x̄ = ∑Mifi/∑fi

Where,

xi –  ith observation

Mi – the midpoint of the ith interval

fi –  corresponding frequency 

n – sample size

2. Median: The median may be defined as the most central observation obtained by sorting the data in ascending order. The formulas for calculating the median are as follows:

• 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

• Grouped data Median: l + [((n / 2) – c) / f] × h

Where,

l – lower limit of the median class given by n / 2

c -cumulative frequency

f – frequency of the median class 

h – class height

3. Mode: The mode is the observation that appears the most frequently in the data collection. The following are the formulae for the mode:

Mode: Most repeated observation (ungrouped data)

Grouped data Mode: L + h[ (fm−f1)/(fm−f1)+(fm−f2) ]

Where,

L – lower limit of given modal class

h – class height

fm – frequency of the modal class

f1 –  frequency of the class preceding the modal class 

f2 – frequency of the class succeeding the modal class

Dispersion Measures:

Measures of dispersion are used in descriptive statistics to identify how spread out a distribution is in relation to the central value. The following are the most significant metrics of dispersion:

Range: The difference between the greatest and lowest value is what the range is defined as. The following is the formula:

Range = H – S

In a data collection, H represents the highest value and S represents the lowest value.

Variance: The variance of a distribution is the variability of the distribution in relation to the mean. The formulae for calculating variance are as follows:

Grouped Data Sample Variance, s2 = ∑f(Mi−X—)2/N−1

Grouped Data Population Variance, σ2 = ∑f(Mi− X—)2/N

Ungrouped Data Sample Variance, s2 = ∑(Xi− X—)2/N-1

Ungrouped Data Population Variance, σ2 = ∑(Xi− X—)2/N

Here,

X¯ – mean

Mi – the midpoint of the ith interval

Xi – ith data point

N – summation of all frequencies

n – number of observations

Standard Deviation: The standard deviation is calculated by taking the square root of the variance. When contrasted to variance, it aids in more effective analysis of a data set’s variability. The following is the formula:

Standard Deviation: S.D. = √Variance = σ

Mean Deviation: The average of the absolute value of the data regarding the mean, median, or mode is called the mean deviation. The absolute deviation is another name for it. The following is the formula:

Mean Deviation = ∑|X−X¯|n

Here,

 X¯ – central value

Conclusion:

Using quantitative analytic methods, descriptive statistics are used to characterize the characteristics of a sample or population. Measures of central tendency and measures of dispersion are two types of descriptive statistics. Descriptive statistics include measurements such as mean, mode, and standard deviation. Tables, charts, and graphs can be used to graphically depict descriptive statistics data.