I'll start with the easier part: the median. The median is a method of central measurement that focuses on the "middle value" - the number which is above half the values and below the other half of the values. The strength of this method of measurement is that it is not skewed by extreme values. Therefore, changing the value of an extreme value would not change the median in any way.
So, if your daily temperatures are 70, 65, 60, 55, 50, 45, and 40, the median would be 55. If the daily temperatures are 700, 65, 50, 55, 50, 45, and 40, the median is still 55.
On the other hand, the mean is greatly affected by extreme values. The strength of this measure is that it includes all the data, but on the other hand just looking at the mean can give you an inappropriate view of the data. Since the mean is weighted by the number of samples in the data, the effect of changing a value depends on how many samples are used to calculate the data. Assuming the month in question has 30 days, the mean will change by (700-70)/30, or 21 degrees.
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