Final Answer:
The statements accurately describes variance is 4) Variance measures how far a data example is from the mean.
Step-by-step explanation:
Variance is a statistical measure that tells us about the spread of a set of numbers. The more varied or dispersed the numbers are, the higher the variance. So, let's go through each of the provided statements to see which one accurately describes variance:
1) Variance measures the error between predicted values and actual values.
This statement is incorrect. The term usually used for the average of the errors between predicted values and actual values is mean squared error (MSE) in the context of predictions, which is related but not the same as variance.
2) Variance measures the shape of the tails in a distribution relative to the center.
This statement is not accurate. The shape of the tails in a distribution is described by a concept called kurtosis, not variance. Kurtosis indicates how tails of the distribution differ from the tails of the normal distribution.
3) Variance measures how much a variable's distribution differs from a normal distribution.
Again, this statement is incorrect. The concept that measures how much a distribution differs from a normal distribution is skewness for asymmetry and kurtosis for the shape of the tails, as mentioned earlier. Variance does not measure the distribution's shape in this way.
4) Variance measures how far a data example is from the mean.
This statement is close but could use a slight clarification. Variance measures how far a set of numbers (or data points) are spread out from their mean (average value), not just a single data example. In other words, it provides a measure of the dispersion around the mean for the entire dataset.
To calculate variance, we follow a series of steps:
a) Find the mean (average) of the data set.
b) Subtract the mean from each data point and square the result (the squared difference).
c) Find the average of these squared differences. This is the variance.
So, looking at these explanations, we can see that statement 4 is the most accurate, as it alludes to variance being a measure of spread from the mean, with the understanding that it applies to the collective differences of the data points from the mean, not just a single data example.