How to know the range of interval frequency table?

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Note: It is customary to use intervals of equal lengths, so I will pretend your first interval is $(11, 20).$


Once data are summarized using intervals (or 'bins'), some information is lost. So it is not possible to find exact values of various numerical descriptive statistics, such as the mean, median, range, and variance.

Even so, one can get approximations. These usually involve making assumptions that one knows are not strictly true, but hopes may be useful.

(a) One assumption is that all observations in an interval are exactly at the midpoint of the interval. Then we would guess the range of your data to be $55.5 - 15.5 = 40,$ and the sample mean to be $\bar X =39.9444,$ the average of the 18 numbers in the vector $$X = (15.5, 15.5, 25.5, 25.5, 25.5, 35.5, 35.5, 35.5, 35.5, 35.5,\\ 45.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5).$$

Similarly, the sample variance is estimated as $s^2 = 214.3791.$ Because some observations are above their interval midpoints and some are below, we can hope that the estimated mean and variance are not far from the values for the un-binned sample.

(b) Another assumptions is that the observations in an interval are 'equidistantly spaced' (according to some specific interpretation) within the interval. Then the sample range might be approximated as something like $57.75-14 = 43.75.$ Some books show a formula that uses this assumption to try to estimate the sample median. (But you did not ask about it and it is seldom used in practice, so I will skip that.)

(c) What can be said for sure, and without making assumptions, is that the range is between $60-11 = 49$ and $51-20 = 31.$

Caution: Before calculators and computers were in common use, it was customary to put data into bins and use such methods to estimate sample descriptive statistics. Especially for very large datasets, this saved a lot of tedious computation. Nowadays in practice, I think binning is mainly used to make histograms. I suppose the purpose of this problem is to get you thinking about the definition of the sample range, not to indicate good statistical practice.

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Updated on September 25, 2020

Comments

  • shashack
    shashack about 3 years

    Interval Frequency

    10~20 .... 2

    21~30 .... 3

    31~40 .... 5

    41~50 .... 1

    51~60 .... 7

    How to know about the range of the above frequency table? I know the range definition... (Highest value - Lowest value) However, we don't know about the detail from the table. In this case, How to calculate the range..

  • shashack
    shashack about 7 years
    Thanks a lot!! I was wondering about the range, and thougt like you. So, I could assure about the question.