In this video I go over the concept of the median, which is a measure of centrality of a probability density function. It is defined as the value of the random variable that 50% of the sample population is greater than while the other 50% are less than. This can be seen visually as the half of the total area of the probability density function to the left and to the right of the median. On the other hand, the mean (or average) is based on the total sum of the random variable as opposed to strictly the sample size which can be skewed higher or lower than the median. This is a pretty interesting video discussing the differences between the mean and median so make sure to watch this video!
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Applications of Integrals: Probability: Median
In my last video, Probability: Average Values: Example 2, I showed that 37% of people are expected to wait longer than the average or mean waiting time of 5 minutes.
What is interesting is that the average wait doesn't necessarily have to mean 50% of callers wait longer than while 50% wait shorter than the average.
This is because the mean depends on the total time waited by all the callers thus if someone waits very long, this can bring the average up, while not necessarily affecting the percentage of callers waiting that long.
For example, consider the case where 3 people wait 1 minute, while 1 person waits 5 minutes:
Now if that 1 person waited 37 minutes instead of only 5:
Another measure of centrality of a probability density function is the median.
The median is a number m such that half the callers have a waiting time less than m and the other callers have a waiting time longer than m.
In general, the median of a probability density function is the number m such that:
This means that half the area under the graph of the probability density function, f, lies to the right of m and the other half to the left of m.
