Statistics Numerical Measures

Data is organized and summarized either graphically or numerically. Graphical descriptions of data are often used. But, if the given data set is large, constructing a graph becomes tedious. Although we can visualize the shape, centre, and spread of the distribution of the data set from the histogram, we cannot quantify data. We need to find out the numerical measures for describing data.

A statistic is a numerical descriptive measure calculated from sample data whereas a parameter is a numerical descriptive measure of a large population. Generally, the values of parameters are not known. We calculate statistics from the sample data, and based on the data in the samples, make claims about the parameters, which represent the population from the sample data. 

Here, we will illustrate the numerical descriptive measures for sample and population, their computation, their meaning, and their uses.

Numerical Descriptive Measures For Sample

There are three types of numerical descriptive measures in statistics.

• Measure of Central Tendency

• Variability

• Shape

Let's discuss each of them 

Measures of Central Tendency

Measure of central tendency is defined as the number that represents the centre of a set of ordered numerical data. The different measures of central tendency are mean, median, mode, and geometric mean.

Mean

Mean, also known as average, is a measure of the central tendency of a group of values.  Mean, generally refers to the arithmetic mean, as opposed to harmonic mean or geometric mean. The value of the mean is extremely affected by outliers.

To calculate the mean, we take the sum of all the values and divide it by the number of values as shown below.

\[\bar{x} = \frac{x_{1} + x_{2} + x_{3} + … + xn}{n}\]

I mean is calculated from the sample of a population, then it is known as sampling mean, represented as \[(\bar{x})\], whereas population mean is represented as \[(\mu)\].

Median 

Median is a measure of central tendency which distributes the data into two parts, separating the upper half and lower half of data by a value known as the median. The median is affected by the extreme values.

Locating The Median

If the given data is arranged in order, then the median is located at \[\frac{n+1}{2}\] data values.

If the number of values is odd, the median is the middle number whereas if the number of values is even, the medium is the average of the two middle numbers.

Note: \[\frac{n+1}{2}\] is not the value of median but only the position of median in the ranked data.

Mode

Mode is the value that occurs most frequently. It is not affected by extreme values. It is used either for categorical data or numerical data. There may be several modes or no modes.

Geometric Mean

In Mathematics, the term Geometric mean is defined as the average or mean which represents the central tendency or typical value of a set of numbers by using the product of their values in opposition to the arithmetic mean that uses sum). For a collection {x₁, x₂,...xₙ} of a positive real number, the geometric mean is defined as:

GM {x₁, x₂,...xₙ} =\[\sqrt[n]{x_{1}, x_{2}, ...xn}\]

Example:

Find the geometric mean of 2 and 32.

GM ( 2, 32) = \[\sqrt{2.32}\] = \[\sqrt{64}\] = 8

Therefore, the geometric mean of 2 and 32 is 8

Measure of Variations

Variations measure the spread or dispersion of values in a data set. The different parameters of variations are:

• Range

• Interquartile range

• Variance

• Standard Deviation

• Coefficient of Variation

Let us discuss each of the parameters of variation

Range

The difference between the greatest value and the smallest value of a given data set is termed as the range. It is the easiest measure of variation.

Range =  Largest  Value - Smallest Value

It ignores how data is distributed and is also sensitive to outliers.

Interquartile Range

The interquartile range is also the measure of spread or variation, based on splitting a given data set into four quartiles. Quartiles divide the rank-ordered set into 4 equal parts. The values that divide each data are known as the first quartile, second quartile, and third quartile, and are represented as Q1, Q2, and Q3.

First Quartile (Q1) -The first quartile divides the series into 4 equal parts. It is also known as the lower quartile. It divides the series in such a way that 25% of the observations are below it and the remaining 75% are above it. 

Second Quartile (Q2) - The second quartile divides the series into 4 equal parts. It is also known as the median. It divides the series equally. 50% of the observations are below it and the other 50% of the observations are above it.

Third Quartile (Q3)- The third quartile divides the series into 4 equal parts. It is also known as the upper quartile. It divides the series in such a way that 75% of the observations are below it and the remaining 25% of the observations are above it.

Interquartile Range = Q3 - Q1

Variance

Variance is the average (approximately) square deviation of values from the mean.

Sample Variance: S² =\[\frac{\sum_{i=1}^{n}(X - \bar{X})^{2}}{n - 1}\]

Here,

\[\bar{X}\] - Arithmetic mean

n = Sample size

Xi = ith value of the variable X

Standard Deviation

Standard deviation is the most commonly used measure of variation for the samples. It shows variation about the means and has the same unit as the original data. 

Sample Standard Deviation: S = \[\sqrt{\frac{\sum_{i=1}^{n}(X - \bar{X})^{2}}{n - 1}}\]

Coefficient of Variation

The term coefficient of variation is defined as the standard deviation, divided by the mean, and multiplied by 100. It is always calculated in percentages and shows variation relative to the mean. 

The coefficient of variation can be used to compare two or more data sets measured in different units.

CV = \[(\frac{S}{\bar{x}})\] * 100

Measure of Variation Summary

• The more the data are spread out, the larger the range, interquartile range, variance, and standard deviation.

• The lesser the data are spread out, the smaller the range, interquartile range, variance, and standard deviation.

• If there is no variation (all values are the same), then all these measures will be 0.

• None of these measures will ever be negative.

Measure of Shape

The shape of the distribution shows how data is distributed. The measures of shape are symmetric or skewed. 

Left - Skewed 

Mean < Median

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Symmetric

Mean = Median

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Right- Skewed

Mean > Median

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Numerical Descriptive Measures For Population

Numerical descriptive measures described previously are of samples, not population

Numerative descriptive measure, describing a population known as parameters, and are represented by Greek letters.

Important population parameters are population mean, population variance, and population standard deviation.

Population Mean - The population mean is the sum of all the values in the population divided by the size of the population, N.

\[\mu\]=\[\frac{\sum_{i=1}^{N}Xi}{N}\]=\[\frac{X_{1} + X_{2} + X_{3}...XN}{N}\] 

Where,

\[\mu\] - Population Mean

N - Population Size

Xi - ith value of the variable X

Population Variance - The population variance is the average of the square deviation of values from the mean.

\[\sigma^{2}\] = \[\sqrt{\frac{\sum_{i=1}^{N}(Xi - \mu)^{2}}{N}}\]

Where,

\[\mu\] - Population Mean

N - Population Size

Xi - ith value of the variable X

Population Standard Deviation - It is the most commonly used measure of variations and has the same unit as the original data.

Population Standard Deviation : \[\sigma\] = \[\sqrt{\frac{\sum_{i=1}^{N}(Xi - \mu)^{2}}{N}}\]

Where,

\[\mu\] - Population Mean

N- Population size

Xi - ith value of the variable X

Solved Examples

1. Find the mean, median, mode, and range for the data given below.

12 , 17, 12, 13, 12, 14, 13, 21, 12

Solution:

Mean - It is the sum of all the values divided by the number of values as shown below.

Mean = \[\frac{12+17+12+13+12+14+13+21+12}{9}\] = 14

Median - The median is the middle or central value of the data set. To calculate the median, we will arrange the data in ascending order as 12, 12, 12, 12, 13, 13, 14, 17, 21.

There are 9 numbers, so the middle value is

\[\frac{9+1}{2}\]= 5

= 5th number

Therefore, the median is 13

Mode - The value that occurs most frequently in a given data is termed as mode. Accordingly, 12 is the mode.

Range - Largest Value - Smallest Value 

Largest Value = 21

Smallest Value = 12

Range = 21 - 12 = 9

2. Find the coefficient of variation for the data given below.

Stock A 

• Average Price of Last Year = 60

• Standard Deviation = 6

Stock B

• Average Price of Last Year = 100

• Standard Deviation = 6

Solution:

Stock A :

• Average Price of Last Year = 60

• Standard Deviation = 6

CV of stock A = \[(\frac{S}{\bar{X}})\] \[\times\] 100% = \[\frac{6}{60}\] \[\times\] 100% = 10%

CV of stock B = \[(\frac{S}{\bar{X}})\] \[\times\] 100% =  \[\frac{6}{100}\] \[\times\] 100% = 6%

Both stock A and stock B have a similar standard deviation, but stock B is less variable in comparison to its price.