Characteristic Compressive Strength
The compressive strength of the concrete is given in terms of characteristic compressive strength of 150mm sized cubes that is tested at 28 days ( fck). The characteristic compressive strength is defined as the strength of the concrete below which not more than 5% of the test results are expected to fall. The compressive strength of the cube samples although exhibit variations, when plotted on a histogram are found to follow the shape of a bell shape curve, as shown below.

The above curve is called the normal or the Gaussian Distribution. The curve is an idealized distribution of the values of compressive strength for a sizeable number of test cubes. The horizontal axis represents the values of the compressive strength. The vertical axis represents the number of test samples for a particular compressive strength. This is also termed as the frequency. The average of the values of the compressive strength i.e. the mean strength is represented by fcm.
Target Mean Strength
The characteristic compressive strength fck, is the value in the Xaxis below 5% of the total area under the curve falls. As shown in the figure1, the value of fck is lower than fcm by an amount equal to 1.65𝛔. Where 𝛔 is the standard deviation of the normal distribution. Therefore
fcm = fck + 1.65𝛔
Where, fcm = Target mean strength, fck = compressive strength, 𝛔 = standard deviation
Therefore for designing concrete mix of a specified characteristic compressive strength, the design strength or the target strength is calculated by the above formula.
For example, the designing of M20 concrete mix, the design strength is fcm = 20 +1.65𝛔
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Concrete Mix Design