Averages and Spread

An Estimate of Central Tendency is a single value used to represent a typical value within the data set. The three most common are the mean, median, and mode.
The mean is calculated by summing all the data values and then dividing by the total number of values.
The median is the middle value in an ordered data set. When the data has an odd number of values, the median is the single middle value. When there’s an even number of values, it’s the mean of the two middle values.
The mode is the value or values that appear most frequently in a data set.
A Measure of Spread describes how much variability there is in a data set, with common examples being the range, interquartile range (IQR), and standard deviation.
The range is calculated by subtracting the smallest value in the set from the largest value.
The interquartile range (IQR) measures the spread of the middle 50% of data or the range of the values between the first quartile and the third quartile.
The standard deviation is a measure of how spread out numbers are from the mean, with a higher standard deviation indicating greater variability.

The mean is typically used when you want to determine the average of a data set, making it key in situations like calculating average grades or earnings.
The median is often used in skewed distributions or when outliers may skew the mean, as it reflects the central value of the data.
The mode could be used when the most frequent occurrence within the data needs to be identified.
The range offers a quick glance at the spread of values but can be greatly influenced by outliers.
The interquartile range (IQR) gives a better sense of the spread around the middle of the data set and is less affected by outliers.
The standard deviation can describe how closely the values cluster around the mean in a more precise way than the range or IQR.

To calculate the mean, sum up all the values and then divide by the count of values.
To find the median, order the values from least to greatest and locate the value in the middle.
The mode is simply the value or values appearing most frequently.
To calculate the range, subtract the smallest value from the largest value.
The interquartile range (IQR) is found subtracting the value of the first quartile from the value of the third quartile.
Calculation of the standard deviation involves several steps: first, calculate the mean; then, subtract the mean from each value to get the deviation of each; next, square these deviations and find their mean; finally, take the square root of this result.

Note that the mean is affected by every single value—the extremely high and low values can significantly shift it.
The median might not always be a value within the original data set, especially when there’s an even number of values.
Be aware that a data set might have more than one mode, or potentially no mode at all if all values occur only once.
While the range might suggest little spread if the value is small, it does not consider if there are values that are significantly far from the rest.
The interquartile range (IQR) provides a more resistant measure of spread, less sensitive to extreme values.
Standard deviation is the most comprehensive measure of spread, providing an understanding of how much the values deviate from the mean on average.