a sample of four observation has a mean of 6, a median of 4, and a mode of 3.find the standard deviation.

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The mode of 3 tells us that in the dataset of 4, three values are the same: (3x+y)/4=mean, 6. The median is the centre value of the ordered values or, if there are an even number of data, the average of the two central values. Whether y is bigger or smaller than x is irrelevant because the average of the central values has to be 4. There must be two 4's at the centre because the mode of 3 forces two 4's into the central position. Therefore, x=4. So (12+y)/4=6, making 12+y=24 and y=12. The standard deviation can now be calculated: subtract the mean from each datum and square it and add these numbers together: 4+4+4+36 = 48 [3*((4-6)^2)+(12-6)^2] and divide by 3 [the number in the dataset less 1]. That gives us 48/3=16, the variance, so the standard deviation is sqrt(16)=4.

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