Quality Assurance/Quality Control (QA/QC) for Waste Stabilization/Solidification - Stabilization and Solidification of Hazardous, Radioactive, and Mixed Wastes - page 323

Average strength

Specified
characteristic
strength

5 percent
defectives

1.64 σ

20 25 30 35 40 45 50 55 60
Compressive strength (MPa)

FIGURE 12.1 Normal distribution of concrete strengths.

It is now generally accepted that the strength of concrete has a normal distribution
as shown in Figure 12.1. A normal distribution is identified by its mean and standard
deviation. The average of all individual tests (X bar) can be calculated using Equation
12.1:

XX ++ + ....... X

1 2 n

(12.1)

X =

n

where X 1 , X 2 , ….., X n are the results of individual tests and n is the total number of
tests made.
The standard deviation is calculated from the same testing set using Equation
12.2.

n

∑ (

2

XX − )

i

i − 1

(12.2)

σ =

n − 1

Other parameters sometimes used are:

σ
X

Coefficient of variation V =

x 100%

Range R = highest value - lowest value

As shown in Figure 12.1, there is always a probability that a strength obtained
in one or more tests is less than the specified strength. It is usual to specify the
quality of concrete not as a minimum strength, but as a “characteristic strength”
below which a specified portion of the test results, often called “defectives,” may
be expected to fall. Due to the variation of concrete in production, it is necessary
to design the mix to have an average strength greater than the specified strength by
an amount termed the margin (k.σ):