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Calculate the mean value and variance of the following data, and identify and discard any possible outliers:

1.73, 1.53, 1.67, 1.69, 1.58, 1.71

Answer :

The mean value and variance of the given data set is 1.65 and 0.0056,there are no outliers in the given data set.

To calculate the mean value of a set of data, you need to add up all the numbers in the set and then divide the sum by the total number of values. Let's calculate the mean value for the given data set: 1.73, 1.53, 1.67, 1.69, 1.58, and 1.71.
Step 1: Add up all the numbers in the data set:
1.73 + 1.53 + 1.67 + 1.69 + 1.58 + 1.71 = 9.91
Step 2: Divide the sum by the total number of values:
9.91 ÷ 6 = 1.65
So, the mean value of the given data set is 1.65.
To calculate the variance of a data set, you need to find the average of the squared differences between each data point and the mean. Let's calculate the variance for the given data set.
Step 1: Find the squared difference between each data point and the mean:
(1.73 - 1.65)^2 = 0.0064
(1.53 - 1.65)^2 = 0.0144
(1.67 - 1.65)^2 = 0.0004
(1.69 - 1.65)^2 = 0.0016
(1.58 - 1.65)^2 = 0.0049
(1.71 - 1.65)^2 = 0.0036
Step 2: Find the average of the squared differences:
(0.0064 + 0.0144 + 0.0004 + 0.0016 + 0.0049 + 0.0036) ÷ 6 = 0.0056167
So, the variance of the given data set is approximately 0.0056.
To identify and discard any possible outliers, you can use the concept of z-scores. A z-score measures how many standard deviations a data point is away from the mean. Typically, any data point with a z-score greater than 3 or less than -3 can be considered an outlier.
To calculate the z-score for each data point, you can use the formula:
z = (x - mean) / standard deviation
In this case, we only have the mean value and variance, so we can calculate the standard deviation by taking the square root of the variance.
Step 1: Calculate the standard deviation:
√0.0056 ≈ 0.0748
Step 2: Calculate the z-score for each data point:
z1 = (1.73 - 1.65) / 0.0748 ≈ 1.07
z2 = (1.53 - 1.65) / 0.0748 ≈ -1.61
z3 = (1.67 - 1.65) / 0.0748 ≈ 0.27
z4 = (1.69 - 1.65) / 0.0748 ≈ 0.54
z5 = (1.58 - 1.65) / 0.0748 ≈ -0.94
z6 = (1.71 - 1.65) / 0.0748 ≈ 0.80
Step 3: Identify any data point with a z-score greater than 3 or less than -3:
None of the calculated z-scores are greater than 3 or less than -3.
Therefore, there are no outliers in the given data set.

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