Does the mean or median better describe the shape of this data set?
46, 54, 43, 57, 50, 62, 78, 42
Both the mean and median can be useful in different ways to describe the shape of a data set.
Mean:
To find the mean of the data set, add up all the numbers and divide by the total number of values:
(46 + 54 + 43 + 57 + 50 + 62 + 78 + 42) / 8 = 54.5
The mean is 54.5. This value can be useful in describing the central tendency of the data set - it gives an idea of the "typical" value. However, if there are any extreme values in the set (either very large or very small), the mean can be easily affected by them, and may not give a good idea of the overall shape of the data.
Median:
To find the median of the data set, put the values in order from smallest to largest:
42, 43, 46, 50, 54, 57, 62, 78
The median is the middle value. If there are an odd number of values, like in this set, it's easy to pick out the middle one:
42, 43, 46, 50, [54], 57, 62, 78
The median is 54. The median is useful in describing the central tendency of the data set as well - in this case, half the values are above 54 and half are below. The median is also more resistant to extreme values than the mean - even if there were some very large or small values in the set, it would not affect the median as much as the mean.
In conclusion, both the mean and median can be useful in describing the shape of a data set. If the values are relatively evenly distributed, the mean and median may be close to each other, and both can give a good idea of the central tendency. However, if the data is skewed in one direction or the other, the median may be a more useful measure of central tendency.