The concept of standard deviation is one of the main concepts of statistics. It is commonly abbreviated as SD and denoted as σ “sigma”. The standard deviation calculates how much individual data points deviate from the mean.
The theory of standard deviation was first proposed by an English prominent mathematician Karl Pearson in the late 19th century. The Standard Deviation Index (SDI) is used by many external diagnostics and clinical laboratories and is calculated as follows:
In this article, we are going to explore the concept of standard deviation in detail. We’ll also discuss the SD formula and will teach the complete procedure to determine SD from grouped and ungrouped data.
What is the Standard Deviation?
The standard deviation in descriptive statistics is a measure of the amount of variation or dispersion in a dataset. It is used to determine the variance or spread by which individual data points deviate from the mean. A small value indicates that the data point is closer to the mean and the value within the data set is relatively consistent.
On the other hand, a higher value shows that the values deviate further from the average. Data values are more diverse, and extreme values are more likely to occur. The standard deviation is of two types and each type has its formula to calculate the value in the dataset.
The SD Formula for Population data,
Where:
- xi = individual data points
- μ = population mean value
- σ = symbol of the Population standard deviation
- n = number of observations
- Σ = sum of a total list of numbers
The SD Formula for Sample Data,
Where:
- s = shown as the Sample standard deviation
- Σ = sum of a total list of numbers
- xi = an individual data point
- x̄ = sample mean value
- n = number of observations
What is the process to Calculate Standard Deviation?
In general, we’re talking about SD so it means we’re talking about population SD. The population standard deviation can be calculated by using the following steps:
- At first, to find the variance we need to calculate the Mean Value = Sum of Obs / No. of Obs
x̄ = Σx / n
- Determine squared differences of the data value from the mean value.
Σ (X – x̄)2
- Calculate the average Squared difference value which is equal to σ2 and known as variance value = Sum of Squ. Diff / of Obs
σ2 = Σ (X – x̄)2 / n
- Take the square root to the resultant variance value.
σ = √ Σ (X – x̄)2 / n
You can also get help from STD calculators to calculate standard deviation online according to the above provided results.
Standard Deviation of Discrete Grouped Data:
For discretely grouped data, we first created a frequency table and then performed further calculations. We can calculate the standard deviation using three different methods:
Actual Mean Method
If the given dataset has n values (x1, x2, x3, …, xn) and their corresponding frequencies are (f1 + f2 + f3 + … + fn) Then the standard deviation can also be calculated by using this formula.
σ = √ Σin fi(xi – x̄)2 / n
Where:
- fi = the total number of frequency (n = f1 + f2 + f3 + … + fn)
- x̄ = the mean value.
Assumed Mean Method
In grouped data, if the values of the given data are very large then assume any number Z as the mean of the data. Then the deviation of each value is di = xi – Z
So, the formula of SD becomes for the assumed mean method is:
σ = √ [(Σin (fi di)2 / n) – (Σin (fi di) / n)2]
Step Deviation Method
This method of deviation is the next step of the assumed mean method formula, in which we compute deviations on assumed value and it becomes d’ = d/i
where “i” is the common factor of d values.
σ = √ [(Σ(f d’)2/ n) – (Σ(f d’)/ n)2] x i
Standard Deviation of Ungrouped Data:
We can also calculate the standard deviation using the same methods as above but in different ways. They are discussing below:
Actual Mean Method
For this method, first, we compute the mean value and then find the deviation of the data. And then the SD formula for the actual mean method is:
σ = √ Σ (X – x̄)2 / n
Assumed Mean Method
If the given data contains large values, then we assume an arbitrary mean value of the data by ourselves Z. Then the deviation of the assumed arbitrary mean value di = xi – Z
so, the SD formula for the assumed mean method we get:
σ = √ [(Σ (di)2 /n) – (Σ (di) /n)2]
Step Deviation Method
As same the above method, also an arbitrary value is assumed by us as the mean value of the data. In this step, we calculate the deviation of the assumed value. Then it became d’ = d/i, where “i” is the common factor of the assumed value.
So, the SD formula for the step deviation method we get:
σ = √ [(Σ(d’)2/n) – (Σ(d’)/n)2] x i
Examples:
Evaluate the standard deviation of the following given data:
Solution:
Step 1 – Find the Mean value.
n = Σf = 70
Mean = x̄ = Σ f.xi / n = 1330/70 = 19
Step 2 – Calculate the squared differences of data value from the mean value = Σ (xi – x̄)2 = [(5+15+25+35+45) – 19(5)]2 = 302 = 900
Step 3 – Calculate the variance = σ2 = Σ (X – x̄)2 / n = 900/70 = 12.857
Step 4 – Take square root of the variance = σ = √Σ (X – x̄)2 / n = √12.857
The Standard Deviation of the given dataset = σ = 3.5856
Inferences:
This article contains the most familiar information on the concept of standard deviation in statistics. We went through the process of calculating SD step by step which is the best and most basic way to find SD and we learned to calculate both (discrete grouped data and ungrouped data) terms that are most important.
Finally, a solved example helps to understand the above discussion. We are sure that after reading this article you will be able to answer any question about standard deviation.
