# Standard Deviation for Compressive Strength of Concrete with Example [PDF]

Standard deviation for concrete is the method to determine the reliability between the compressive strength results of a concrete batch. The standard deviation serves as the basis for control of variability in the test results of concrete for the same batch of concrete.

It is a statistical method that is based on the correlation analysis, testing of hypothesis, analysis of variance, and regression analysis to compare two or more series of compressive strength of concrete concerning their variability.

In simple words, the standard deviation manifests the range of dispersion or variation in the result that exists from the mean, average, or expectedÂ value.

## Calculation of Standard Deviation for Concrete

The calculation of standard deviation for compressive strength of concrete can done in 2 ways:

### 1. Assumed Standard Deviation

The minimum number of cube test samples required to derive the standard deviation is 30. In the case, where sufficient test results for a particular grade of concrete are not available, the value of standard deviation is assumed as per the IS-456 Table 8 (ClausesÂ 3.2.1.2) as shown below:

Table 1: Assumed Standard Deviation

 Sl.No Grade of Concrete Characteristic compressive strength (N/mm2) Assumed standard deviation (N/mm2) 1 M10 10 3.5 2 M15 15 3 M20 20 4 4 M25 25 5 M30 30 6 6 M35 35 7 M40 40 8 M45 45 9 M50 50 10 M55 55

However, as soon as the minimum number of test results are available, the derived standard deviation shall be calculated and used.

Note - The above values are dependent on site-control- having proper storage of cement,Â weigh batching of all materials, controlled addition of water, regular checking of all elements such as aggregate grading and moisture content, and regular checking of workability and strength.

### 2. Derived Standard Deviation

When the number of test results available are more than 30, the standard deviation of the test results is derived by the following method -

Where,
phi = Standard DeviationÂ
Âµ = Average Strength of Concrete
n = Number of Samples
x = Crushing value of concrete in N/mm2

The value of standard deviation will be lesser if the quality control at the site is excellent, and most of the test results will be approximately equal to the mean value. If quality control is unsatisfactory, then the test results will have much difference from the mean value, andÂ therefore, the standard deviation will be higher.

The permissible deviation in the mean of compressive strength of the concrete is as per the below table prescribed by IS-456 Table No-11.

Table 2: Characteristic Compressive Strength Compliance Requirement

## Example Calculation of Standard Deviation for M60 grade Concrete with 33 cubes.

A concrete slab of 400Cum was poured for which 33 cubes were cast for 28 days compressive test. The standard deviation for the 33 number of cubes tests is calculated below-

Table 3: Test Result of Concrete Cubes

 SL No Weight of the cube in Kg Max Load in KN Density in Kg/Cum Compressive Strength in Mpa Remarks 1 8.626 1366 3594.2 60.71 Pass 2 8.724 1543 3635.0 68.57 Pass 3 8.942 1795 3725.8 79.77 Pass 4 8.850 1646 3687.5 73.15 Pass 5 8.466 1226 3527.5 54.48 Fail 6 8.752 1291 3646.7 57.37 Fail 7 8.806 1457 3669.2 64.75 Pass 8 8.606 1285 3585.8 57.11 Fail 9 8.708 1465 3628.3 64.71 Pass 10 8.696 1387 3623.3 61.64 Pass 11 8.848 1476 3686.7 65.60 Pass 12 8.752 1529 3646.7 67.95 Pass 13 8.450 1564 3520.8 69.51 Pass 14 8.708 1703 3628.3 75.68 Pass 15 8.602 1478 3584.2 65.68 Pass 16 8.762 1539 3650.8 68.40 Pass 17 8.468 1475 3528.3 65.55 Pass 18 8.862 1386 3692.5 61.60 Pass 19 8.728 1507 3636.7 66.97 Pass 20 8.480 1550 3533.3 68.88 Pass 21 8.708 1738 3628.3 77.24 Pass 22 8.712 1463 3630.0 65.02 Pass 23 8.562 1327 3567.5 58.97 Fail 24 8.370 1529 3487.5 67.99 Pass 25 8.592 1388 3580.0 61.68 Pass 26 8.622 1383 3592.5 61.46 Pass 27 8.732 1245 3638.3 55.39 Fail 28 8.776 1482 3656.7 65.86 Pass 29 8.724 1367 3635.0 60.75 Pass 30 8.628 1590 3595.0 70.66 Pass 31 8.604 1394.7 3585.0 61.98 Pass 32 8.566 1406.1 3569.2 62.49 Pass 33 8.578 1387.2 3574.2 61.65 Pass Â Â Â Total 2149.22 Â Â Â Â Average 65.12 Â

Table 4: Calculation of Standard Deviation

Sum of (x-Âµ)2 =1132.55
SD = SqRt (1132.55/(33-1))

Standard Deviation = 5.94 N/mm2

As per the IS-456, for concrete of grade above M-20,

1. fck + 0.825 x derived standard deviation
= 60+0.825*5.94
=64.90 N/mm2
2. fck + 4 N/mm2
=60+4
=64 N/mm2

The highest value if the above two is considered, which is
Standard Deviation= 64.90 N/mm2

From table-3, we have the average/mean value of the compressive strength which is 65.12N/mm2, which is higher than the standard deviation 64.90N/mm2

## Conclusion

From Table-3, it can be noticed that the test results of five cubes are below 60 N/mm2, which means the cubes have failed. But from the standard deviation calculation, the concrete member can be approved, and non-destructive tests are not prescribed.

1. What is Standard Deviation for Compressive Strength of Concrete?

The standard deviation of concrete is the reliability between the different compressive strength results of a concrete batch. It is also defined as the range of dispersion or variation in the compressive strength result that exists from the mean, average, or expectedÂ value.

2. What is the importance of Standard Deviation for concrete?

The standard deviation of concrete accounts the deviations in the compressive strength results due to poor handling of concrete used while storing, mixing, transportation, and testing of concrete.