Approximately normal distributions are so common that we allude to them in every

Question

Approximately normal distributions are so common that we allude to them in everyday language without being aware of it. For example, we often assume that by having told someone the average of a set of values we have communicated something useful. Or we may say that most of the outcomes of a process are similar, meaning that their values exhibit little dispersion/variability. Indicate how becoming comfortable with the use of formal central tendency descriptors (mean, median, & mode) and formal dispersion/variability descriptors (range & standard deviation) might improve communication in cases where more than casual understanding is important.

Explanation / Answer

The important thing to consider with the measures of central tendency and dispersion/ variability is that within themselves, they portray or convey slightly different meanings to the data set of study. In other words, the mean vs median vs mode values, which are all central tendency measures, will translate to slightly different interpretations for the nature of values.

To makw this point more clear, let us say we are given that for some data set whose values go from 0 to 100, mean is 50, median is 65 and mode is 46. This implies that while the average is 50, more values are bunched towards the right i.e. the distribution is positively skewed, thus giving us a median of 65. Such a subtle interpretation cannot be conveyed by a casual understanding. Similarly, the mode is 46 would imply that this particular value occurs most often, which has no relation to the mean or median.

Again, the range and standard deviation have different connotations to given data set. The range on a very broad level tells how high and low the values can go. If there are any outliers, that directly impacts range. Whereas standard deviation is slightly less impacted, though not completely unaffected by outliers in the sense that S.D. considers the root mean square difference of the values from the mean. Hence it is a much better indicator of the dispersion behaviour. A range of 100 in a data set with a minimum value of 0 simply tells that the highest value is 100, which does not tell us much about how the values are distributed. If we are told that S.D. is 5, that means values are quite closely clustered around the mean. Whereas if S.D. had been 25, we would conclude that the values vary a lot.

Hence to summarize, a casual understanding of data indicators will just be that. We shall not je able to draw these fine line interpretations which can only come with a deeper knowledge.

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