10x.4. Properties

Both variance and SD are always ≥ 0.

The first property of both variance and standard deviation is that they cannot be negative. This makes sense because both metrics are derived from squared differences from the mean, and squaring any number (whether positive or negative) always results in a non-negative value. Therefore, the smallest value variance or standard deviation can take is zero, which occurs only when all data points are identical, meaning there is no variability or spread among the data.

Think of variance and standard deviation like a measure of how far apart points are on a race track. If all runners finish at the same spot, there’s no distance between them; hence, the measure of spread (variance or SD) is zero. You can never have a negative distance on the track!

Both increase if data has extreme values.

The presence of outliers—data points that are significantly higher or lower than the rest of the dataset—tend to increase both variance and standard deviation. This happens because outliers contribute to a larger squared difference from the mean. For example, if in a set of test scores, most students score between 60 and 70, but one student scores 20, this 20 drastically increases the average squared deviation, thereby increasing the overall variance and standard deviation.

Imagine you are measuring friends' heights, and they range from about 150 cm to 180 cm, but one friend is 210 cm tall. This tall friend skews the average height significantly, similar to how an outlier affects variance and standard deviation by making the spread appear larger than it normally would.

Var(𝑋+𝑌) = Var(𝑋)+ Var(𝑌)

This property highlights that if you have two independent random variables, the variance of their sum is equal to the sum of their variances. This means when combining two sets of data (like scores from different exams), you can calculate the variance of the total data simply by adding the variances of each dataset together, assuming the variables do not influence each other.

Think of combining two jars of marbles, each representing some data set. The variance of the mixed jar is just the sum of the variances of both jars because you can think of the different colors of marbles as independent data observations that don’t affect each other’s contribution to the overall variability.

If 𝑌 = 𝑎𝑋+𝑏, then Var(𝑌)= 𝑎² ⋅Var(𝑋) and SD(𝑌)= |𝑎|⋅SD(𝑋)

This rule states that if you linearly transform a dataset (multiply by a constant 'a' and add a constant 'b'), the variance of the new dataset (Y) can be expressed in terms of the variance of the original dataset (X) multiplied by the square of the scaling factor, while the standard deviation is multiplied by the absolute value of the scaling factor. This property is crucial in engineering applications where measurements and their uncertainties are subject to transformation.

Consider blowing up a balloon. As you inflate it (scaling), the distance between any two points on its surface expands as well. If the original distance (variance) between points was small, scaling up that distance by a factor results in a larger distance ‘spread’ on the balloon’s surface, mirroring how variance scales with a transformation.

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