# Computing the Standard Deviation

Understanding standard deviation is not just about grasping its conceptual meaning, but also about knowing how to compute it in practice. The difference between low and high variability in a dataset can be best appreciated when observed through tangible examples.

In the following sections, we’ll walk through two computational examples — one showcasing a dataset with low variability and another with high variability. By analyzing these examples side-by-side, we aim to offer a clearer perspective on how the standard deviation, as a measure of spread, reflects the dispersion of scores around the mean in various scenarios. Let’s dive in.

## Computational Example

Let’s delve into a hands-on example to solidify our understanding.

### Low Variability

Student Quiz Score (X) Deviation (X – Xbar) Squared Deviation (X – Xbar)^2
A 85 0 0
B 86 1 1
C 84 -1 1
D 85 0 0
E 85 0 0
F 86 1 1
Sum 511 1 3
• Sample Mean (Xbar) = 511/6 = 85.17 (rounded to two decimal places)
• Sum of Squared Deviations = 3
• Variance = 3/5 = 0.6
• Sample Standard Deviation (s) = sqrt(0.6) = 0.77 (rounded to two decimal places)

### High Variability

Student Quiz Score (X) Deviation (X – Xbar) Squared Deviation (X – Xbar)^2
A 70 -15 225
B 90 5 25
C 60 -25 625
D 100 15 225
E 75 -10 100
F 95 10 100
Sum 490 -20 1300
• Sample Mean (Xbar) = 490/6 = 81.67 (rounded to two decimal places)
• Sum of Squared Deviations = 1300
• Variance = 1300/5 = 260
• Sample Standard Deviation (s) = sqrt(260) = 16.12 (rounded to two decimal places)

## Discussion

The standard deviation (SD) offers insights into the dispersion or spread of scores around the mean. In our examples:

• For the low variability dataset, the scores are tightly packed around the mean with an SD of only 0.77. This indicates that most students scored very close to the mean score.
• For the high variability dataset, there’s a more considerable spread around the mean with an SD of 16.12. This means there’s a wider range of scores, and students’ performance varies significantly.

Low variability (small SD) suggests consistency, uniformity, or little deviation in the data. It’s like saying most members of a group are very similar in the trait being measured. On the other hand, high variability (large SD) indicates greater diversity or discrepancy in the data, implying that members of the group can be quite different from one another in the measured trait.

By comparing the SDs in both examples, one can easily discern that the first set of scores is more consistent and has less variability, while the second set of scores exhibits a broader range of performance.

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`Last Modified:  10/16/2023`

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