Differences between Range and Standard Deviation

Difference between std::set::upper_bound and std::upper_bound in C++

In statistics, Range and standard deviation provide insight into the spread or dispersion of data points within the data set. Range and standard deviation are both measures of variability in a dataset, but they differ in their calculation and interpretation.

The purpose of this article is to know the difference between range and standard deviation for the students offering clarity on calculations.

What is Range?

Range is the difference between two extreme observations of the distribution or data. It provides a measure of the dispersion or spread of the data, indicating the extent to which the values vary from each other. If A and B are the greatest and smallest values observed respectively in a data, then its range is A-B.

Thus,

Range = maximum value – minimum value

What is Standard deviation?

Standard deviation is defined as the measure of all data variation from its mean value or the positive square root of the variance X is known as standard deviation. It provides information about how much individual data points deviate from the mean value of the dataset.

Mathematically, the standard deviation (σ) is calculated using the following formula:

Standard deviation (σ) = √VAR(X)

Where:

Var(X) determines the variatnce of x.

Difference between Range and Standard deviation

The key difference between range and standard deviation is given below:

Range	Standard Deviation
Measures the difference between the highest and lowest values of the distribution.	Measures the dispersion of data points around the mean
Simple to calculate and understand.	Requires more computational effort and statistical knowledge.
Susceptible to outliers.	Less affected by outliers.
Useful for a quick overview of data.	Offers a more understanding of data variability.
Range is used in exploratory data analysis.	Standard deviation is used in statistical analysis, finance and quality control.
Consider only two extreme data’s.	Consider every point in the dataset.

Read More,

Solved Examples on Range and Standard Deviation

Example 1: Calculate the Range and standard deviation for the following dataset: 10,15,20,25,30.

Solution:

Range = (maximum value- minimum value)

( 30-10) = 20.

For Standard deviation following steps are used

Calculate Mean

We need to calculate the mean of the dataset before finding standard deviation,

Mean = (10+15+20+25+30)/5 = 20

Calculate the Deviations from the Mean

Deviation from the mean for each value = Value – Mean

Deviations: (-10), (-5), 0, 5, 10

Calculate the Squared Deviations

Squared deviation for each value = (Deviation from the mean)²

Squared deviations: 100, 25, 0, 25, 100

Calculate the Variance

Variance = (Sum of squared deviations) / (Number of values) = (100 + 25 + 0 + 25 + 100) / 5 = 250 / 5 = 50

Calculate the Standard Deviation

Standard deviation = √variance = √50

=7.07

Example 2: Consider the following set of numbers representing the daily temperatures (in degrees Celsius) for a week: 20, 22, 24, 23, 25, 21, 19.

Solution:

Arrange the numbers in ascending order: 19, 20, 21, 22, 23, 24, 25.

Range = Largest value – Smallest value = 25 – 19 = 6.

So, the range of the daily temperatures for the week is 6 degrees Celsius.

For Standard deviation following steps are used

Calculate Mean

Mean = (19 + 20 + 21 + 22 + 23 + 24 + 25) / 7 = 154 / 7 ≈ 22.

Calculate the Deviations from the Mean

Deviation from the mean for each temperature = Temperature – Mean

Deviations: -2, 0, 2, 1, 3, -1, -3

Calculate the Squared Deviations

Squared deviation for each temperature = (Deviation from the mean)²

Squared deviations: 4, 0, 4, 1, 9, 1, 9

Calculate the Variance

Variance = (Sum of squared deviations) / (Number of temperatures) = (4 + 0 + 4 + 1 + 9 + 1 + 9) / 7 = 28 / 7 = 4

Calculate the Standard Deviation

Standard deviation = √(Variance) = √4 = 2

So, the standard deviation of the daily temperatures for the week is approximately 2 degrees Celsius.

Conclusion

Both range and standard deviation offer insights into data variability. While both the range and standard deviation provide measures of variability, the standard deviation is often preferred for its ability to capture the overall dispersion of data points around the mean.

FAQs on Range and Standard Deviation

How do I calculate Range and standard deviation?

The Range is calculated by subtracting the lowest value from the highest value. The Standard deviation involves several steps:

Calculate the mean

Finding the squared differences between each data point and the mean.

Taking the square root of the average of these squared differences.

What is the significance of understanding Range and Standard deviation?

Understanding these measures helps in comprehending the spread of data, aiding decision making process in various fields like finance, research and quality control.

Can the range be negative?

No, the range cannot be negative because it is the difference between the highest and the lowest values in a dataset.

How is the range calculated?

To calculate the range, subtract the minimum value from the maximum value in the dataset.

Which is more sensitive to outliers, range, or standard deviation?

The range is more sensitive to outliers because it depends only on the extreme values in the dataset. Standard deviation is less sensitive to outliers because it considers the deviation of each data point from the mean.

How is standard deviation calculated?

The standard deviation is calculated by taking the square root of the average of the squared differences between each data point and the mean.

Tags:

#Maths-Difference-Between #Mathematics #School Learning