# FAQ: Statistical Thinking - Outliers and Robust Measures

This community-built FAQ covers the “Outliers and Robust Measures” exercise from the lesson “Statistical Thinking”.

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Hey there!

I am a little confused about the movement of the median in the “McCartney, BTS, and Beyonce move to the city” example. How does this example produce the calculation that median income moves from \$32,978 to \$33,011, which is a \$33. I am trying to reconcile that with the addition of 3 individual data points.

Thanks!

Suppose we have five data points

``````2, 3, 6, 7, 8
``````

For the mean, we just add up all the values and then divide by the number of data points,

``````Mean = (26 / 5) = 5.2
``````

For the median, we arrange the values in ascending order. If there are an odd number of data points, then we just pick the middle value. If there are an even number of data points, then there is no single data point right in the middle. So we usually take the average of the middle two values.

``````# (Five data points in total, so middle value is third data point)
Median = 6
``````

Suppose, we add four more data points as outliers, so that our data is now:

``````2, 3, 6, 7, 8, 15, 32, 87, 90

Mean = (250 / 9) = 27.8

# (Nine data points in total, so middle value is fifth data point)
Median = 8
``````

The outliers changed the mean significantly from `5.2` to `27.8`, but the median didn’t change as noticeably from `6` to `8`.

In the example mentioned by you, we haven’t been explicitly given all the data points. So, let’s just pretend that there are `1001` data points,

``````# Pretend data
..., \$32900, \$32978, \$33000, \$33022, ... , \$81516

# Mean = ( (sum of all 1001 incomes) / 1001 )
Mean = \$34795

# Suppose \$32978 is the 501st data point
# (500 data points --- middle data point --- 500 more data points),
Median = \$32978
``````

Now, let’s add three outliers so that our data has `1004` data points:

``````# Pretend data
..., \$32900, \$32978, \$33000, \$33022, ... , \$81516, \$48000000, \$57000000, \$81000000

# Mean = ( (sum of all 1004 incomes) / 1004 )
Mean = \$228235

# Median
# (501 data points --- 502nd data point, 503rd data point --- 501 more data points)
# Since there are an even number of observations (1004), so the median will
# be the average of the 502nd and 503rd data point,
Median = (\$33000 + \$33022) / 2
Median = \$33011
``````

Since the lesson hasn’t given us the data points, so the above is just a pretend example.

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