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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/mm^{2}) |
Assumed standard deviation (N/mm^{2}) |

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/mm^{2}

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

Specified Grade | Mean of Group of 4 Non-Overlapping Consecutive test results in N/mm^{2} |
Individual Test Results in N/mm^{2} |

M-15 | f_{ck} + 0.825 x derived standard deviationor f _{ck }+ 3 N/mm^{2}(Whichever is greater) |
Greater than or equal to -Â f_{ck}^{-3} N/mm^{2} |

M-20 and above | f_{ck} + 0.825 x derived standard deviationor f _{ck }+ 4 N/mm^{2}(Whichever is greater) |
Greater than or equal to -Â f_{ck}^{-4} N/mm^{2} |

## 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/mm ^{2}**

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

- f
_{ck}+ 0.825 x derived standard deviation

= 60+0.825*5.94**=64.90 N/mm**^{2} - f
_{ck }+ 4 N/mm^{2}

=60+4**=64 N/mm**^{2}

The highest value if the above two is considered, which is

Standard Deviation= **64.90 N/mm ^{2}**

From table-3, we have the average/mean value of the compressive strength which is **65.12N/mm ^{2}**, which is higher than the standard deviation

**64.90N/mm**

^{2}## Conclusion

From Table-3, it can be noticed that the test results of five cubes are below 60 N/mm^{2}, 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.

**Read More: ****1. ****Compressive Strength of Concrete -Cube Test [PDF], Procedure, Results**

2. What is Ultrasonic Testing of Concrete for Compressive Strength?

3. Concrete Compressive Strength Variation with Time