How To Calculate Standard Deviation In Excel Manually
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Calculating Standard Deviation in Excel Manually
Calculating standard deviation is a fundamental task in statistics, allowing you to understand the spread or dispersion of a dataset around its mean. While Excel offers built-in functions for direct standard deviation calculation, understanding the manual process provides a deeper insight into the concept and reinforces your grasp of statistical principles. This guide will walk you through calculating standard deviation in Excel, step-by-step, without using built-in functions like `STDEV.S` or `STDEV.P`. We will cover both sample and population standard deviation.
Understanding Standard Deviation
Before diving into the calculations, let’s briefly review what standard deviation represents. * **Standard Deviation:** A measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (average) of the set, while a high standard deviation indicates that the values are spread out over a wider range. * **Sample Standard Deviation (s):** Estimates the standard deviation of a population based on a sample drawn from that population. It uses `n-1` in the denominator for a more accurate estimate of the population standard deviation, especially with smaller sample sizes. * **Population Standard Deviation (σ):** Measures the standard deviation of an entire population. It uses `n` in the denominator.
Steps to Calculate Standard Deviation Manually in Excel
Let’s assume you have a dataset in Excel, with each value in a separate cell. We will calculate both sample and population standard deviation. For our example, let’s use the following data, entered into cells A1 through A5: | Cell | Value | |—|—| | A1 | 10 | | A2 | 12 | | A3 | 15 | | A4 | 18 | | A5 | 20 |
1. Calculate the Mean (Average)
The first step in calculating standard deviation is to find the mean (average) of the dataset. This is the sum of all the values divided by the number of values. * **Formula in Excel:** In cell A6 (or any empty cell), enter the following formula to calculate the mean: `=SUM(A1:A5)/COUNT(A1:A5)` * `SUM(A1:A5)` calculates the sum of the values in cells A1 through A5 (10 + 12 + 15 + 18 + 20 = 75). * `COUNT(A1:A5)` counts the number of values in cells A1 through A5 (5). * The formula divides the sum by the count, giving you the mean (75 / 5 = 15). Cell A6 will now display the value 15.
2. Calculate the Deviations from the Mean
Next, you need to calculate the deviation of each value from the mean. This is simply the difference between each value and the mean. * **Formula in Excel:** In cell B1, enter the following formula: `=A1-$A$6` * `A1` refers to the first value in your dataset (10). * `$A$6` refers to the mean we calculated in step 1 (15). The dollar signs (`$`) create an absolute reference, which means that when you copy the formula down, it will always refer to cell A6. * This formula calculates the difference between the first value and the mean (10 – 15 = -5). * **Copy the Formula:** Click on cell B1, then drag the small square at the bottom-right corner of the cell down to cell B5. This will copy the formula to cells B2, B3, B4, and B5, automatically adjusting the `A1` reference to `A2`, `A3`, `A4`, and `A5`, respectively. The deviations will be calculated for each value in your dataset. Column B will now show: | Cell | Value | |—|—| | B1 | -5 | | B2 | -3 | | B3 | 0 | | B4 | 3 | | B5 | 5 |
3. Square the Deviations
To eliminate negative values and emphasize larger deviations, square each of the deviations calculated in the previous step. * **Formula in Excel:** In cell C1, enter the following formula: `=B1^2` * `B1` refers to the deviation calculated in step 2 (-5). * `^2` raises the deviation to the power of 2 (squares it). * **Copy the Formula:** Click on cell C1, then drag the small square at the bottom-right corner of the cell down to cell C5. This will copy the formula to cells C2, C3, C4, and C5, automatically adjusting the `B1` reference. The squared deviations will be calculated for each value. Column C will now show: | Cell | Value | |—|—| | C1 | 25 | | C2 | 9 | | C3 | 0 | | C4 | 9 | | C5 | 25 |
4. Calculate the Sum of Squared Deviations
Add up all the squared deviations you calculated in the previous step. * **Formula in Excel:** In cell C6 (or any empty cell), enter the following formula: `=SUM(C1:C5)` * `SUM(C1:C5)` calculates the sum of the values in cells C1 through C5 (25 + 9 + 0 + 9 + 25 = 68). Cell C6 will now display the value 68.
5. Calculate the Variance
The variance is the average of the squared deviations. It differs slightly depending on whether you’re calculating the sample variance or the population variance. * **Sample Variance:** Divide the sum of squared deviations by `n-1`, where `n` is the number of values in your dataset. * **Formula in Excel (Sample Variance):** In cell D1 (or any empty cell), enter the following formula: `=C6/(COUNT(A1:A5)-1)` * `C6` refers to the sum of squared deviations (68). * `COUNT(A1:A5)` counts the number of values (5). * `-1` subtracts 1 from the count (5 – 1 = 4). * The formula divides the sum of squared deviations by `n-1` (68 / 4 = 17). Cell D1 will now display the sample variance, 17. * **Population Variance:** Divide the sum of squared deviations by `n`, where `n` is the number of values in your dataset. * **Formula in Excel (Population Variance):** In cell E1 (or any empty cell), enter the following formula: `=C6/COUNT(A1:A5)` * `C6` refers to the sum of squared deviations (68). * `COUNT(A1:A5)` counts the number of values (5). * The formula divides the sum of squared deviations by `n` (68 / 5 = 13.6). Cell E1 will now display the population variance, 13.6.
6. Calculate the Standard Deviation
Finally, calculate the standard deviation by taking the square root of the variance. Again, this differs slightly for sample and population. * **Sample Standard Deviation:** Take the square root of the sample variance. * **Formula in Excel (Sample Standard Deviation):** In cell D2 (or any empty cell), enter the following formula: `=SQRT(D1)` * `D1` refers to the sample variance (17). * `SQRT()` is the Excel function for calculating the square root. * The formula calculates the square root of the sample variance (√17 ≈ 4.123). Cell D2 will now display the sample standard deviation, approximately 4.123. * **Population Standard Deviation:** Take the square root of the population variance. * **Formula in Excel (Population Standard Deviation):** In cell E2 (or any empty cell), enter the following formula: `=SQRT(E1)` * `E1` refers to the population variance (13.6). * `SQRT()` is the Excel function for calculating the square root. * The formula calculates the square root of the population variance (√13.6 ≈ 3.688). Cell E2 will now display the population standard deviation, approximately 3.688.
Summary of Formulas
| Calculation | Formula | |—|—| | Mean | `=SUM(A1:A5)/COUNT(A1:A5)` | | Deviation | `=A1-$A$6` (copied down) | | Squared Deviation | `=B1^2` (copied down) | | Sum of Squared Deviations | `=SUM(C1:C5)` | | Sample Variance | `=C6/(COUNT(A1:A5)-1)` | | Population Variance | `=C6/COUNT(A1:A5)` | | Sample Standard Deviation | `=SQRT(D1)` | | Population Standard Deviation | `=SQRT(E1)` | By following these steps, you can manually calculate the standard deviation in Excel, solidifying your understanding of this important statistical measure. While Excel’s built-in functions offer a quicker approach, this manual method provides valuable insight into the underlying calculations.
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