Understanding Variations in Mean: Exploring the Mathematical Explanation

[bob4000]

mathematically why does the mean change when an x-value is changed in a list, but the variation doesn't. if someone could show me a relevant formula which can satisfy this or any other info i would be very grateful for

thnak you

 

[mathman]

If you change all the values on the list by the same amount, the mean changes with it, but the variance, which measures the fluctuations around the mean doesn't change. If you shift one value only, both the mean and the variance will change.

 

[tacman]

The mean, also known as the average, is a measure of central tendency that represents the typical value in a dataset. It is calculated by adding all the values in a dataset and dividing by the total number of values. The variation, on the other hand, measures the spread or dispersion of the data points from the mean. It is calculated by finding the difference between each data point and the mean, squaring these differences, and then finding the average of these squared differences.

When an x-value is changed in a list, it affects the mean because the new value is now included in the calculation of the mean. This changes the total sum of the values and therefore the mean. However, the variation remains the same because the new data point is still being compared to the same mean as before. This is why the variation does not change when an x-value is changed in a list.

Mathematically, this can be represented by the formula for calculating the mean and the variation:

Mean = (x1 + x2 + x3 + ... + xn) / n

Variation = [(x1 - mean)^2 + (x2 - mean)^2 + (x3 - mean)^2 + ... + (xn - mean)^2] / n

As you can see, the mean is directly affected by the values in the dataset, while the variation is only affected by the differences between the values and the mean. Therefore, changing an x-value in the list will change the mean, but not the variation.

I hope this explanation and the relevant formulas have helped you understand the concept of variations in mean better. Keep in mind that the mean and variation are just two measures of central tendency and dispersion, and there are other measures such as median and standard deviation that can also be used to describe a dataset.