Standard deviation is one of the values that we need when calculating the target mean strength of concrete. However, many examples give the standard deviation directly without explaining how it was calculated.
In this article, we will calculate it step by step from concrete compressive strength results. Then, we will use it in the required average strength calculation.
What Is Standard Deviation of Concrete Strength?
Simply, standard deviation shows how much the concrete strength test results are spread around their average value.
If the standard deviation is small, the test results are close to each other and the concrete production is more consistent. On the other hand, a large standard deviation means that the results are more scattered.
This variation may come from changes in materials, water-cementitious materials ratio, batching, mixing, sampling, curing, specimen preparation, or testing.
Standard Deviation Formula
| Symbol | Meaning |
|---|---|
| S | Sample standard deviation |
| xi | Each concrete strength test result |
| x̄ | Average or mean of all test results |
| n | Number of strength tests |
| Σ | Sum of the values |
We divide by n – 1, not n, because the strength results are a sample used to estimate the variation of the concrete production. This is also why you should use STDEV.S in Excel.
Before Starting the Calculation
One strength test is not always one cylinder. For ACI concrete strength evaluation, one strength test is the average strength of companion cylinders made from the same concrete sample and tested at the same age.
Therefore, enter the strength test result in the calculation, not every cylinder result separately. Always check the definition in the code and project specification you are following.
Worked Example Using 15 Concrete Strength Tests
Assume we have the following 15 consecutive 28-day concrete compressive strength test results. The specified concrete strength is 30 MPa.
The following results will be used to explain the complete calculation.
Step 1: Calculate the Mean Strength
First, add all the strength test results and divide the total by the number of tests.
Step 2: Calculate the Difference from the Mean
For each result, subtract the mean strength. Then, square the difference. We square the differences because positive and negative values would otherwise cancel each other.
| Test | Strength, xi (MPa) | xi – x̄ | (xi – x̄)2 |
|---|---|---|---|
| 1 | 31 | -1.20 | 1.44 |
| 2 | 32 | -0.20 | 0.04 |
| 3 | 33 | 0.80 | 0.64 |
| 4 | 30 | -2.20 | 4.84 |
| 5 | 34 | 1.80 | 3.24 |
| 6 | 32 | -0.20 | 0.04 |
| 7 | 31 | -1.20 | 1.44 |
| 8 | 35 | 2.80 | 7.84 |
| 9 | 33 | 0.80 | 0.64 |
| 10 | 32 | -0.20 | 0.04 |
| 11 | 34 | 1.80 | 3.24 |
| 12 | 31 | -1.20 | 1.44 |
| 13 | 30 | -2.20 | 4.84 |
| 14 | 33 | 0.80 | 0.64 |
| 15 | 32 | -0.20 | 0.04 |
| Total | 483 | 0.00 | 30.40 |
Step 3: Calculate the Standard Deviation
We have 15 tests. Therefore, n – 1 = 14.
S = sqrt(2.1714)
S = 1.47 MPa
So, the standard deviation of the 15 concrete strength tests is 1.47 MPa.
Modification Factor When Fewer Than 30 Tests Are Available
The calculated value is not always the final value used for concrete proportioning. When only 15 to 29 consecutive tests are available, multiply the calculated standard deviation by a modification factor.
| Number of tests | Modification factor |
|---|---|
| Less than 15 | Do not use this method |
| 15 | 1.16 |
| 20 | 1.08 |
| 25 | 1.03 |
| 30 or more | 1.00 |
For intermediate numbers of tests, interpolate between the listed factors.
In our example, we have 15 tests:
The data should represent the same materials, proportions, and concreting conditions proposed for the new work. The specified strength should be within 7 MPa of the strength represented by the test data, and the data should cover at least 45 days.
Using the Standard Deviation to Calculate Required Average Strength
Now, let us use the modified standard deviation in the required average strength calculation. For this example, f’c = 30 MPa and S = 1.71 MPa.
For concrete with f’c not greater than 35 MPa, calculate the following two values and use the larger result. Check the current edition of the code and the project specification before using them for an actual concrete mix.
| Check | Calculation | Result |
|---|---|---|
| Equation 1 | f’cr = 30 + (1.34 x 1.71) | 32.29 MPa |
| Equation 2 | f’cr = 30 + (2.33 x 1.71) – 3.45 | 30.53 MPa |
| Adopted value | Use the larger result | 32.29 MPa |
Therefore, the required average compressive strength is 32.29 MPa.
Example 2: Plain Concrete Pavement
Assume we have a plain concrete pavement. The specified compressive strength is 35 MPa and the concrete producer’s standard deviation is 2.0 MPa.
The two strength checks are:
f’cr = 35 + (2.33 x 2.0) – 3.45 = 36.2 MPa
The larger value is used. Thus, the required average compressive strength is 37.7 MPa.
How to Calculate It in Excel
Enter the 15 strength test results in cells B2 to B16.
| Calculation | Excel formula | Result |
|---|---|---|
| Mean strength | =AVERAGE(B2:B16) |
32.20 MPa |
| Sample standard deviation | =STDEV.S(B2:B16) |
1.47 MPa |
| Modified standard deviation for 15 tests | =STDEV.S(B2:B16)*1.16 |
1.71 MPa |
Do not use STDEV.P for this calculation. STDEV.P divides by n, while the sample standard deviation used here divides by n – 1.
Concrete Strength Standard Deviation Calculator
Enter the strength test results separated by commas, spaces, or new lines. Use one value for each strength test, not one value for each companion cylinder.
Common Mistakes
- Using every cylinder result separately when the code defines one test as the average of companion specimens.
- Dividing by n instead of n - 1.
- Using results from different concrete mixtures or different production conditions in one calculation.
- Forgetting the modification factor when only 15 to 29 tests are available.
- Using an old standard deviation without checking whether the materials and production conditions are still representative.
- Using the calculated standard deviation without checking the current code and project specification.
What If Fewer Than 15 Tests Are Available?
Do not force a standard deviation calculation from a very small record and use it for concrete proportioning. When enough data are not available, use the required average strength values or the trial-mixture procedure given in the applicable code.
Therefore, if fewer than 15 eligible tests are available, follow the no-data or trial-mixture procedure in the applicable code instead of using an unreliable value.
Summary
To calculate the standard deviation of concrete compressive strength, first calculate the mean strength. Then, find the difference between each result and the mean, square the differences, add them, divide by n - 1, and take the square root.
For the 15-test example, the calculated standard deviation is 1.47 MPa. After applying the 1.16 modification factor, the value used in the required average strength calculation becomes 1.71 MPa.
References
- Kosmatka, S. H., and Wilson, M. L., Design and Control of Concrete Mixtures, 16th edition, Portland Cement Association, 2016, Chapter 15, pp. 365-366 and 379.
- ACI 214R-11, Guide to Evaluation of Strength Test Results of Concrete (reapproved 2019).
- NIST/SEMATECH e-Handbook of Statistical Methods, Measures of Scale.
Read Also:
Calculation of Concrete Target Mean Strength
Characteristic Compressive Strength vs Target Mean Strength