Mean, Median, Mode and Histograms
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flagWhat you'll discover
- arrow_forwardCompute the mean, median, mode and range of a raw data set
- arrow_forwardExplain what each average tells you and when it misleads
- arrow_forwardGroup data into classes and draw a histogram
- arrow_forwardRead the modal class off a histogram
- arrow_forwardDecide which average is the right one for a given situation
The three averages
The mean is the sum of all the values divided by how many there are: x̄ = Σx/n. It uses every value, so a single extreme value drags it.
The median is the middle value when the data is arranged in order. With an even count it is the mean of the two middle values. The median is not affected by how extreme the extremes are, which is why house prices and exam marks are often reported as medians.
The mode is the value that appears most often. A data set can have no mode (nothing repeats) or several modes. The range is the largest value minus the smallest, and it measures how spread out the data is.
Grouped data and the histogram
When there are many readings you group them into classes of equal width and count the frequency in each class. A histogram has the classes on the x-axis and the frequency on the y-axis, and the bars touch, because the classes are continuous.
The class with the tallest bar is the modal class, and it is the only kind of "mode" that exists for grouped data. For grouped data the mean is computed from the mid-values: x̄ = Σfm/Σf, where m is the mid-value of each class.
The choice of class width changes how the histogram looks but not the underlying data — that is worth saying in an answer that asks you to compare two groupings.
Choosing the right average
Use the mean when the data is fairly even and you want to use every value: the average height of a class, the average rainfall of a month.
Use the median when a few extreme values would distort the mean: the average income of a neighbourhood, the average waiting time at a hospital. Use the mode when the most common item is the point: the most common shoe size to stock, the most common score in a game.
A question that gives you three averages and asks which to use is testing this judgement, not your arithmetic. Say which you choose and why in one sentence.