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Given a data set as follows:

$$\begin{array}{crrrrrrrr} Classes: & Less\:than\:20 & 20-30 & 30-40 & 40-50 & 50-60 \\ frequency: & 30 & 20 & 15 & 10 & 5 \end{array}$$

The objective is to find a suitable measure of dispersion.

The first thing I thought of was the variance, but can we find out the mid point of a class interval without the lower point ?

Then, I thought of quartile range, but the lower quartile lies in the first class only, so couldn't proceed further.

Next range, again lower and upper limits required.

Can anyone help me with this ? Any suggestions on what measure of dispersion could be used ?

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    $\begingroup$ Selecting one's statistic based on reviewing the data is a dicey procedure. For exploratory analysis it's fine, but for almost anything else it is difficult to justify. Could you explain what this measure of dispersion will be used for? You might also explain how the class cutpoints were selected (do they depend on the data, too?) and provide a lower limit for the open-ended "less than 20" class. $\endgroup$ – whuber May 18 '17 at 13:15
  • $\begingroup$ Is the number of observations in the class "greater than 60" equal to zero or do you have right truncated data, that is, the number of observations in this class is unkown? $\endgroup$ – Jarle Tufto May 18 '17 at 13:59
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The theoretical lower limit of the lowest class is 0 (although the actual limit could be higher). It can't be negative. But you would still need to make some assumptions to get either the standard deviation or the interquartile range. For the former, you'd have to assume something about the distribution in every level; for the latter, about the levels containing the quartile values.

Since the distribution is quite skewed, you might also consider the median absolute deviation - that requires similar assumptions to the sd, but may be a better measure.

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    $\begingroup$ How can we be so sure that the lower limit is zero ? $\endgroup$ – User9523 May 18 '17 at 12:13
  • $\begingroup$ You are right. I edited my answer $\endgroup$ – Peter Flom - Reinstate Monica May 18 '17 at 12:29
  • $\begingroup$ What is the basis for determining this distribution is "quite skewed"? How would one go about finding the MAD, given the open-ended lower class and the fact that computing differences of two already uncertain values is even more uncertain? $\endgroup$ – whuber May 18 '17 at 13:16
  • $\begingroup$ Skewed because the lowest category also has the most kids and the highest has very few. I don't think it's possible to come up with a set of numbers that follow this distribution and are not quite skewed. The lower class isn't open ended - we don't know it's exact bottom, but it can't be below 0. If you assume a distribution within each class, you can calculate SD or MAD, I just suggested MAD would be better than SD $\endgroup$ – Peter Flom - Reinstate Monica May 18 '17 at 17:43

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