The standard deviation for ungrouped data measures the spread or dispersion of a set of data points around their mean. Here's a breakdown of how to calculate it:
Steps to Calculate Standard Deviation for Ungrouped Data
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Calculate the Mean:
- The first step, as highlighted in the reference video, is to calculate the mean (average) of the data set.
- You find the mean by adding all the data points together and then dividing by the total number of data points.
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Calculate the Deviations from the Mean:
- Subtract the mean from each individual data point. This gives you the deviation of each data point from the mean.
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Square the Deviations:
- Square each of the deviations calculated in the previous step. This makes all the values positive and emphasizes larger deviations.
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Calculate the Mean of the Squared Deviations:
- Add up all the squared deviations, and then divide by the total number of data points. This gives you the variance.
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Calculate the Square Root:
- Take the square root of the variance you calculated in the last step. The result is the standard deviation.
Formula Summary
While the steps are described above, the following formulas can be used:
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Mean (μ): μ = (Σx) / n
- Where Σx is the sum of all values, and n is the total number of values.
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Variance (σ²): σ² = Σ(x - μ)² / n
- Where (x-μ) are the deviations from the mean, and n is the total number of values.
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Standard Deviation (σ): σ = √σ²
- Where σ² is the variance.
Example
Let's say we have the following ungrouped data: 4, 8, 6, 5, 3.
| Data Value (x) | Deviation (x - μ) | Squared Deviation (x - μ)² |
|---|---|---|
| 4 | -1.2 | 1.44 |
| 8 | 2.8 | 7.84 |
| 6 | 0.8 | 0.64 |
| 5 | -0.2 | 0.04 |
| 3 | -2.2 | 4.84 |
- Mean (μ): (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
- Variance (σ²): (1.44 + 7.84 + 0.64 + 0.04 + 4.84) / 5 = 14.8 / 5 = 2.96
- Standard Deviation (σ): √2.96 = 1.72 (approximately)
Importance of Standard Deviation
Standard deviation gives you an idea of how spread out the numbers are in your dataset.
- A small standard deviation suggests that data points are clustered closely around the mean.
- A large standard deviation indicates that the data is more dispersed.
This measure is critical in various fields like statistics, finance, and science for data analysis and decision-making.
