The standard deviation, represented by σ or s, is a statistical term that quantifies the dispersion of data in relation to its mean. It is either 0 or constantly positive. The data are more loosely grouped around the mean when the standard deviation is small, whereas a high standard deviation suggests that the data are widely distributed about the mean.
Only when every value in the data is equal and there is no fluctuation in the data does the data have a standard deviation of zero. A single data point's average deviation from the mean is shown by the standard deviation; however, any one observation in your data may deviate more or less from the mean than one standard deviation.
Standard deviation units can be used to express the distance in terms of standard deviation between any point in a data set and the data set mean. The concept of the standard deviation is very useful when dealing with the data values that are distributed normally.
Here in this comprehensive discussion, we will address this concept through its definition and its important uses. We will provide here a step-by-step guide to compute the standard deviation for any set of data values.
What is standard Deviation (σ / s)?
A useful parameter to have when analyzing the spread of the data values for normal distributions is the standard deviation. When data is distributed normally, there is no skew and it is symmetrical. With values dropping off as they move farther from the center, most values cluster around a central spot.
You can find out how far the given data values are on average from the distribution center by studying the standard deviation. The standard deviation for the population is usually represented by the Greek letter σ or sigma.
The lower-case letter s is frequently used to represent the standard deviation when it is determined for a sample. Standard deviation can alternatively be shortened to SD or STDEV. The mathematical formula to compute the SD or STDEV is:
For population STDEV:
σx = √ [∑ (xi - μ) 2 / N]
Where:
- σx is the population standard deviation,
- ∑ is the symbol for summation indicating that you should take the sum of everything that follows it
- xi is a particular value in your data.
- μ is the population mean
- (xi - μ) is the distance between a particular value in your population data and the mean
- N is the population size.
For sample STDEV:
Sx = √ [∑ (xi - x¯) 2 / N - 1]
Where:
- Sx is the sample standard deviation
- (xi - x¯) is the distance between a particular value in your sample data and the mean
- n is the sample size.
Note: When calculating the sample standard deviation, we divide by N - 1 to get a closer approximation of the true population standard deviation σx.
How to Calculate SD?
Observe that the mathematical relations are identical. The denominator only signifies the actual difference. The sample statistic Sx is divided by the sample size less one (N - 1) while concerning the population parameter σ is divided by the population size N.
These formulas may appear complicated at first, but if you examine them carefully and temporarily ignore the square root symbol, you will see that all you are doing is calculating the mean of the data (xi - x¯) 2 in the case of the population and the average of the squared distances between each data point (xi - μ) 2.
If you are unaware of the symbol Σ, it represents a summation. It means take the sum of thus you are adding up all of the squared differences between the data points and the mean and then dividing by the entire number of data points (N) or if the sample standard deviation is involved, by the total number of data points minus one (N - 1).
We will also elaborate in detail to explain how we square these distances first and then find their square root in the example section.
Example Section:
Now we will discuss below solved examples of calculating standard deviation.
Example 1:
Determine the population STDEV and the sample STDEV for the given ungrouped data values in the table.
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9
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17
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14
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19
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16
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15
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13
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8
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12
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7
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Solution:
Step 1: Compute the average of data values given in the above table.
x¯ = μ = (9 + 17 + 14 + 19 + 16 + 15 + 13 + 8 + 12 + 7) / 10
x¯ = μ = (130) / 10
x¯ = μ = 13
Step 2: Now we will perform the necessary core values computations to determine the standard deviation.
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xi
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(xi - x¯) = (xi - μ)
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(xi - x¯)2 = (xi - μ)2
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9
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-4
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16
|
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17
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4
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16
|
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14
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1
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1
|
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19
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6
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36
|
|
16
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3
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9
|
|
15
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2
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4
|
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13
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0
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0
|
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8
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-5
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25
|
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12
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-1
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1
|
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7
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-6
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36
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∑
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144
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Step 3: To compute the population STDEV:
σx = √ [∑ (xi - μ) 2 / N]
σx = √ [(144) / 10]
σx = √ (14.4)
σx = 3.7947
Now, to compute the sample STDEV:
Sx = √ [∑ (xi - x¯) 2 / N - 1]
Sx = √ [(144) / (10 - 1)]
Sx = √ [(144) / 9]
Sx = √ (16)
Sx = 4 Ans.
Example 2:
Determine the population STDEV and the sample STDEV for the given ungrouped data values in the table. These values represent the scores of eight form-two students in a math test as follows:
Solution:
Step 1: Compute the average of data values given in the above table.
x¯ = μ = (45 + 52 + 54 + 55 + 57 + 57 + 62 + 66) / 8
x¯ = μ = (448) / 8
x¯ = μ = 56
Step 2: Now we will perform the necessary core values computations to determine the standard deviation.
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xi
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(xi - x¯) = (xi - μ)
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(xi - x¯)2 = (xi - μ)2
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45
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-11
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121
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52
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-4
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16
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54
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-2
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4
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55
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-1
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1
|
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57
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1
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1
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57
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1
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1
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62
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6
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36
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66
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10
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100
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∑
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280
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Step 3: To compute the population STDEV:
σx = √ [∑ (xi - μ)2 / N]
σx = √ [(280) / 8]
σx = √ (35)
σx = 5.9161 Ans.
Now, to compute the sample STDEV:
Sx = √ [∑ (xi - x¯)2 / N - 1]
Sx = √ [(280) / (8 - 1)]
Sx = √ [(280) / 7]
Sx = √ (40)
Sx = 6.3246 Ans.
Wrap Up:
In this article, we have discussed a step-by-step comprehensive guide to compute the standard deviation for ungrouped data. We have discussed the mathematical formulation for the population standard deviation and the sample standard deviation precisely.
In the last section, we have solved some examples to apprehend the necessary computations. We hope that by reading this article you can determine the standard deviation of both categories easily.
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