How do you find the median from this table?

I know that there's this formula for finding out the median, but in this case, I need help on what the class interval and the lower boundaries are in order to complete the formula.

An example of the table:

Mass (g)

< 24.5

< 34.5

< 44.5

< 54.5

< 64.5

Cumulative Frequency

7

18

35

44

50

This indicates that <24.5 g has a cumulative frequency of 7 and so on.

2 Answers

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  • 9 years ago
    Favorite Answer

    There's not really a "formula" for finding the median of a group of values (assuming you don't have a mathematical model of the distribution). You simply sort them in order from least to greatest, and whichever value is in the middle, that's the median. If there are an odd number of values, say n = 5, that's easy. You order them 1 2 3 4 5, and the middle value is in the (n + 1) / 2 position. In this case, that's (5 + 1) / 2 = 3rd position.

    If you have an even number of values, there isn't a single value in the middle. So, you take the average of the two values on either side. If n = 6, then you order the values 1 2 3 4 5 6, and take the average of the values in the n / 2 and (n / 2) + 1 positions. In this case, 6 / 2 = 3, and (6 / 2) + 1 = 4. So, you'd average the values in the 3rd and 4th positions.

    Let's try that out with your numbers. If I assume you want the median value of the cumulative frequency totals, then we first count the number of values - 5, which is odd. Conveniently, they're already sorted from least to greatest, so we don't have to do that work. Then, we simply take the (5 + 1) / 2 = 3rd value from the list, and that's the median. That value is 35.

    Now, let's say we had 6 values instead of 5, with the last value 52. Now, we have an even number of values, so we take the average of the 6 / 2 = 3rd and (6/2) + 1 = 4th values, which are 35 and 44. The average of those values is (35 + 44) / 2 = 79 / 2 = 39.5. So, in our hypothetical case, the median is 39.5.

    I suspect that, because you're given a cumulative frequency, you might need to first figure out the actual frequency in each group. That is, how many fragments have mass < 24.5 g, and how many fragments have 24.5 g ≤ mass < 34.5 g, how many have 34.5 g ≤ mass < 44.5 g, etc.

    To find that, we simply take away the objects in the smaller group. In other words, the frequency in the group 54.5 g ≤ mass < 64.5 g is equal to the number of objects < 64.5 g, minus the number of objects < 54.5 g. So, that would be 50 - 44 = 6.

    If we do that, we get

    < 24.5 g: 7

    < 34.5 g: 18 - 7 = 11

    < 44.5 g: 35 - 18 = 17

    < 54.5 g: 44 - 35 = 9

    < 64.5 g: 50 - 44 = 6

    So, if we want the median of those, we'll sort from least to greatest,

    6 7 9 11 17

    And we find the median, which is again in the (5 + 1) / 2 = 3rd position, is 9.

    I hope that helps!

  • 9 years ago

    The mean is the usual average. http://www.purplemath.com/modules/meanmode.htm

    Here, however, I assume that each of the numbers in column 1 are to be multiplied by those in column 2, before calculating the median:

    24.5 x 7

    34.5 x 18

    etc.

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