## What are Descriptive Measures?

A mass that has certain general characteristics as a form of a measure dealing with quantitative data is referred to as descriptive measures. The descriptive measure has different types depending on the different characteristics of the data.

The three main types of descriptive statistics are the central tendency, frequency distribution, and variability of a dataset. A distribution corresponds to the frequencies of different types of responses.

The descriptive statistics are partitioned into measures of central tendency and measures of variability. The measures of central tendency are the mean, median, and mode. Whereas variance, standard deviation, minimum and maximum variables, and kurtosis and skewness are the measures of variability or spread.

## What are the Characteristics of Descriptive Measures?

• It summarizes the characteristics of the provided data set.
• It has two basic types of measures: the measure of variability and the measure of central tendency.
• The measures of central tendency reflect how the data revolves around the center of the data.
• The measure of variability, also called the measure of spread, reflects how the data is dispersed in the given set.

## Measures of Central Tendency

A measure of central tendency describes the data set with a single value representing the center of its distribution. The major types of measures of central tendency are as follows:

### Mean

The average of a data set is called the mean of the data set. It is evaluated as the quotient of the sum of all observations by the sample size. The mean is affected by the extreme values thus, it is not a strong statistic. Very large or very small data values can distract the mean from the center of the data.

(a) Arithmetic Mean

Out of all the averages, the most common average is the arithmetic mean. It is evaluated using the formula:

$\mu =\frac{1}{N}\sum _{i=1}^{N}{x}_{i}$

(b) Other Means

The other types of means include the geometric mean, logarithmic mean, and harmonic mean.

The nth root of the product of n observations from a data set is defined as the geometric mean of the set:

Geometric Mean:- $G=\sqrt[n]{{x}_{1}{x}_{2}...{x}_{n}}$

The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer.

Logarithmic Mean =$\frac{\mathrm{ln}{x}_{2}-\mathrm{ln}{x}_{1}}{{x}_{2}-{x}_{1}}$

The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data.

Harmonic Mean =$\frac{1}{\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+...}$

### Median

The middle value of a data set is called the median of the data. The median divides the data set into two halves and is called the 50th percentile. The median is much less affected by the outliers and the skewed data than the mean. If the number of elements in a dataset is odd, then the middlemost element of the data arranged in ascending or descending order is the median. If the number of elements in the dataset is even, the average of the two central elements of the arranged data is the median of the set.

Example 1: Consider the five items: 12, 13, 21, 27, 31. Since, the data is arranged in ascending order and 21 is in the middle, thus 21 is the median.

Example 2: Consider the six items: 12, 13, 21, 27, 31, 33. Since, the data is arranged in ascending order, and 21 and 27 are in the middle, thus the median is (21 + 27) / 2 = 24.

### Mode

The value that occurs most frequently in a dataset is called the mode of the data. If no two categories in a data are the same, then the dataset has no mode. A dataset may have more than one mode, if multiple categories repeat an equal number of times. The mode is the only measure of central tendency that is used for categorical variables.

## Measures of Variability

The measures of variability are also known as the spread of the data. These measures describe how similar or varied is the set of observations. The most frequently used measures of variability are the range, interquartile range (IQR), standard deviation, and variance.

### Range

The difference between the largest and smallest observations of a data set is known as the range of the set. The bigger the range, the more spread out is the data.

### IQR

The measure of statistical dispersion between the upper (75th) quartiles, i.e. Q3, and the lower (25th) quartiles, i.e. Q1, is called the interquartile range (IQR) of the data. This can be understood by the below example.

The difference between the range and IQR is that while the range measures where the beginning and end of the data point are, the interquartile range is a measure of where the majority of the values lie.

### Variance

The average squared deviation from the mean is called the variance. The variance is evaluated by finding the difference between every data point and the mean, squaring all these values, and then taking the average of those squared numbers.

The problem with variance is that it is not in the same unit of measurement as the original data. This problem occurs due to the squaring of the deviations.

Variance: ${\sigma }^{2}=\frac{\sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{2}}{N}$, where $\mu$ denotes the mean.

The variance of sample data is evaluated as ${s}^{2}=\frac{\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{2}}{N-1}$, where $\overline{x}$ denotes the sample mean.

### Standard Deviation

The square root of the variance is called the standard deviation. It is used more often because it is in the original unit as the data.

Standard deviation: $\sigma =\sqrt{\frac{\sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{2}}{N}}$, where $\mu$ denotes the mean.

The SD of sample data is evaluated as $s=\sqrt{\frac{\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{2}}{N-1}}$, where $\overline{x}$ denotes the sample mean.

A low standard deviation means that the data points tend to be close to the mean, whereas a high standard deviation means that the data points are spread out over a wide range.

It is best used in the case of unimodal data. In the case of a normal distribution, the standard distribution is approximately 34% away from the mean, i.e., 34% of the data values lie between the mean ad the standard deviation. Since a normal distribution is symmetrical, about 68% of the data points fall between one standard deviation above and one standard deviation below the mean. In the same way, approximately 95% of the data values lie between two standard deviations below the mean and two standard deviations above the mean, and approximately 99.7% of data values lie between three standard deviations above and three standard deviations below the mean.

The picture below illustrates that perfectly.

The “Z-Score” is used to check how many standard deviations below (or above) the mean a specific data point lies.

## Examples of Measures of Dispersion

The measures of central tendency look at the average or middle values of a data set, whereas the measures of dispersion look at the spread or variation of the data. The variation refers to the amount by which the values vary among themselves. The values in a data set that are relatively close to each other have a lower measure of variation, and the values that are spread farther apart have higher measures of variation.

Examine the two histograms below.

Both the groups have the same mean weight of 267 lb., but the values of Group A are more spread out compared to the values in Group B, i.e., the weights of Group A are more variable.

## Practice Problems

### Example1

Suppose there is a seesaw and three people. The weights of the three people are 140 lb., 150 lb., and 180 lb. respectively. The people are arranged along with the seesaw according to their weight, i.e., the lightest is at one end, the heaviest at the other, while the third person is in between according to his weight.

We wish to know where to put the fulcrum so that the seesaw is just balanced. The average weight is given by :

$\frac{140+150+180}{3}=156.67$

Therefore, if the fulcrum is set at 156.67, the seesaw will just balance.

### Example 2

Find the nean for the following sample data set: 6.4, 5.2, 7.9, 3.4

Answer:

$\overline{x}=\frac{6.4+5.2+7.9+3.4}{4}=5.725$

### Example 3

Evaluate the median of the following data set: 31, 44, 28, 51, 23, 47, 27, 39, 40, 42, 35.

Answer: To calculate the median with an odd number of values (n is odd), arrange the data in ascending order.

The arranged data is: 23, 27, 28, 31, 35, 39, 40, 42, 44, 47, 51.

The median is 39.

### Example 4

Evaluate the median of the following data set: 31, 44, 28, 23, 47, 27, 39, 40, 42, 35.

Answer: To calculate the median with an even number of values (n is even), first arrange the data in ascending order and take the average of the two middle values.

The arranged data is: 23, 28, 29, 31, 35, 39, 40, 42, 44, 47.

The central values are 35 and 39, so the median is:

$M=\frac{35+39}{2}=37$

### Example 5

Evaluate the variance of the sample data: 3, 5, 7 where the sample mean is 5.

Answer:

${s}^{2}=\frac{{\left(3-5\right)}^{2}+{\left(5-5\right)}^{2}+{\left(7-5\right)}^{2}}{3-1}=4$

### Example 6

Evaluate the range for the given data set.

12, 29, 32, 34, 38, 49, 57

Range = 57 – 12 = 45

## Formulas

 Arithmetic Mean $\mu =\frac{1}{N}\sum _{i=1}^{N}{x}_{i}$ Geometric Mean $G=\sqrt[n]{{x}_{1}{x}_{2}.....{x}_{n}}$ Logarithmic mean Harmonic Mean Variance Standard Deviation Sample Variance Sample Standard Deviation

## Common Mistakes

• Calculation mistakes
• Use of wrong formulae
• Error in writing the formula
• Substituting wrong values
• Error in converting less than and more than data types to standard frequency tables
• Forget to compute the cumulative frequency

## Context and Applications

Descriptive statistics is useful for:

• School & college-level students
• Post-graduation courses in mathematics
• Engineering courses
• Data analysis courses
• Statistics
• Data analysis
• Number systems
• Measurements

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