How to Find Standard Deviation

Step-by-step Guide to Find Standard Deviation

Here is a step-by-step guide to finding standard deviation:

Step 1: Preliminary Considerations

1. Data Comprehension: Familiarize yourself with the data set. Does it encompass the entire population or merely a sample?
2. Nomenclature Familiarity: For the entire population, use ‘\(σ\)’ (sigma) as the standard deviation. For a sample, ‘\(s\)’ denotes the standard deviation.

Step 2: The Process

A. Data Aggregation and Initial Preparation:

1. Collate the Data: Ensure that all data points are in a singular, comprehensible list or column.
2. Count the Entries: Number your data points. Let’s name this number ‘\(N\)’ for a population or ‘\(n\)’ for a sample.

B. Calculating the Mean (Average):

1. Summation: Add up all the individual data points. Let’s denote this sum as \(Σx\).
2. Division: Divide \(Σx\) by ‘\(N\)’ (or ‘\(n\)’ for a sample). This gives you the mean (often symbolized as “\(μ\)” for population mean or “\(x̄\)” for sample mean).
   • The Equation for Mean:
     • For Population: \(μ =\frac{Σx}{N}\)
     • For Sample: \(x̄ = \frac{Σx}{n}\)

C. Determining Deviations from the Mean:

1. Deduction: Subtract the mean (\(μ\) or \(x̄\)) from each individual data point. This will yield the deviation of each data point from the mean.
2. Tabulation: For clarity, you might wish to create a new list or column that represents these deviations.

D. Squaring the Deviations:

1. Amplification of Deviance: Square each of the deviations acquired in the previous step. This is vital as squaring ensures all values are positive, emphasizing the magnitude over direction.
2. New Column Creation: For enhanced clarity and organization, pen down these squared deviations in yet another column or list.

E. Summation of Squared Deviations:

1. Aggregate: Sum up all squared deviations. This aggregated value can be termed as the “sum of squares.”

F. Division by Count:

1. Population Versus Sample:
   • For Population: Divide the sum of squares by ‘\(N\)’.
   • For Sample: Divide by ‘\(n-1\)’. This act, known as Bessel’s correction, counteracts the potential bias in estimating population variance from a sample.

G. Extraction of the Root:

1. Final Touch: Take the square root of the result obtained from the previous step. The outcome of this act is the standard deviation (σ for population, s for sample).

Congratulations, ardent seeker of knowledge! You’ve now delved deep into the heart of data variation and emerged with the standard deviation in tow. This newfound understanding will undeniably augment your prowess in statistical analysis and data interpretation. Remember, the standard deviation is not just a mere statistic, but a story of spread, variability, and the essence of your data’s heartbeats.

Example 1: Daily Number of Coffee Cups Consumed

Suppose you’ve been tracking the number of coffee cups you consume every day for a week. Your data for the \(7\) days is as follows:

Data Set: \(2, 3, 1, 2, 4, 2, 3\)

Step 1: Compute the Mean:

\(x̄\)\(=(2+3+1+2+4+2+3)÷7=17÷7=2.43\)

Step 2: Compute Deviations from the Mean:

\((2−2.43),(3−2.43),(1−2.43),(2−2.43),(4−2.43),(2−2.43),(3−2.43)=−0.43,0.57,−1.43,−0.43,1.57,−0.43,0.57\)

Step 3: Square the Deviations:

\(0.1849,0.3249,2.0449,0.1849,2.4649,0.1849,0.32490\)

Step 4: Sum of the Squared Deviations:

\(5.7143\)

Step 5: Divide by Count (since it’s a sample, \(n-1\)):

\(Variance(s^2)=5.7143÷7=0.8163\)

Step 6: Extract the Root for Standard Deviation:

\(s=\sqrt{0.8163}=0.9035\)

So, the standard deviation for your coffee consumption is approximately \(0.9035\) cups.

Example 2: Scores of Students in a Mini Quiz

Imagine a mini-quiz was conducted in a class, and five students scored as follows:

Data Set: \(8, 10, 9, 7, 6\)

Step 1: Compute the Mean:

\(x̄\)\(=(8+10+9+7+6)÷5=40÷5=8\)

Step 2: Compute Deviations from the Mean:

\((8−8),(10−8),(9−8),(7−8),(6−8)=0,2,1,−1,−2\)

Step 3: Square the Deviations:

\(0,4,1,1,4\)

Step 4: Sum of the Squared Deviations:

\(10\)

Step 5: Divide by Count (since it’s a sample, \(n-1\)):\

\(Variance(s^2)=10÷4=2.5\)

Step 6: Extract the Root for Standard Deviation:

\(s=\sqrt{2.5}=1.58\)

Thus, the standard deviation for the student scores is approximately \(1.58\) points.