Question

For the next three problems, consider the following sample of python numbers per lake in the Florida Everglades {1, 5, 6, 7, 9, 9, 11, 13, 16}

1. Find the mean , median and mode of the above sample .

2. Compute the standard deviation and the variance of the above sample

3.Determine quantitatively if there any unusual values in the above sample ? explain

Answer 1

Solution :

1) The sample mean is given as follows :

Where, xi's are sample values and n is sample size.

We have, n = 9

The mean is 8.556.

Median :

Number of observations (n) = 9, which is an odd number. Therefore, median = ((n + 1)/2)th observations of sorted data.

Our data is already in ascending order, so we don't need to sort the data.

Median = (9+1)/2 = 5th observation

5th observation = 9

Hence, median = 9

Mode : Mode is that value in the data set, which occurs most frequently in the data.

Since, in the given sample data, 9 occurs most frequently, therefore 9 will be the mode.

Mode = 9

2) The variance of the sample data is given as follows :

Where, S² is variance of sample data, xi's are sample values, n is sample size and x̄ is sample mean.

We have, n = 9 and  x̄ = 8.556

The variance of the sample is 20.0278.

The standard deviation of the sample data is given as follows :

The standard deviation of the sample is 4.4752.

3) A value of the sample is said to be unusual if it is more than 2 standard deviations below the mean (i.e. value < (x̄ - 2S)) or it is more than 2 standard deviations above the mean (i.e. value > (x̄ + 2S)).

(x̄ - 2S) = 8.556 - (2 × 4.4752) = -0.3944

(x̄ + 2S) = 8.556 + (2 × 4.4752) = 17.5064

All the sample values are between -0.3944 and 17.5064.

Since, all the sample values are within 2 standard deviations about the mean, therefore we can say that there is no unusual value is the sample.