(i) It does not recognize the individual items, From the very definition of statistics given by Prof. Horace Sacrist “By statistics we mean aggregates of facts… and placed in relation to each other” it is clear that statistics refers to only aggregates of items and that it does not recognize an individual or isolated item which is not placed in relation to any other item. Thus, individual figures like death of five passengers in a bus accident, B.Com. results of a college in a particular session, population of a country in a particular year will not constitute statistics unless they are placed in a group of similar nature. In the words of W.I.King, “Statistics for their very nature of subject can not and will never be able to take into account individual cases. When these are important, other means must be used for their study.” Thus, statistics does not deal with the individual items howsoever important they may be, but with aggregates, although, in the process of analysis through statistical methods the aggregates, although, in the process of analysis through statistical methods the aggregates are reduced to individual figures like that of average, coefficient, index number etc.
(ii) It is not concerned with the qualitative phenomena. From all the definitions of statistics given in the plural sense it is clear that statistics refers only to those data which are numerically expressed. From this it follows that any information which is not numerically expressed nor capable of being expressed numerically will not amount to statistics howsoever it may satisfy the other characteristics of statistics. Thus, qualitative phenomena like honesty, sincerity, greatness, intelligence, efficiency, blindness, deafness etc. can not be studied statistically, The reason behind this is that statistics deals with the process of counting, measurement and weighing etc. while the qualitative data like honesty, integrity, sincerity, intelligence etc. can not be measured, weighed or counted. However, if these phenomena could be transformed to some sort of quantitative form they can be studied statistically without any difficulty. For example, intelligence of a group of students can be studied statistically after assigning the intelligent quotients to each of them through a test of I.Qs. Similarly, honesty of a society can be studied statistically with reference to the comparative figures of crimes committed over a number of years. If the number of crimes in a particular year is found to be less than those in the other years it can be concluded that there has been an improvement in the concerned with the qualitative phenomena.
However, pure qualitative data can not be studied statistically. Hence, it is said that statistics is not concerned with the qualitative phenomena.
(iii) It does not reveal the entire story of a phenomenon. Most of the phenomena are affected by a number of factors all of which are not capable of being expressed quantitatively. Hence, it is not possible to examine a problem statistically in all its manifestations and arrive at the correct conclusions. Many phenomena are to be examined in the light of many things viz : conditions of life, education, culture, religion, philosophy, administration, etc. which can not be studied statistically. Thus, a particular aspects expressed numerically may not be correct and reliable as it is not drawn on the entire background of the phenomenon. Thus, it is said that statistics does not reveal the entire story of a phenomena.
(iv) It results are true only on an average. In the words of W.I.King, “Statistics largely deals with the averages and these averages may be made up of individual items radically different from each other.”
From the observation of Mr. King, it is clear that statistical results are not absolutely true. They are not applicable to all individual cases, although they are derived as the average result of all the individuals forming the group. For example, the average marks of a group of students does not mean that each student of the group has secured the same average mark. Statistical results reveal only the average behaviour; they are, therefore, useful only for a general appraisal of a phenomenon and not for any specific unit or event.
(v) Its laws are not exact. We have seen that the two fundamental laws of statics viz. (i) The law of statistical regularity and (ii) The law of inertia of large numbers, are probabilistic in nature. They are not deterministic like those of the physical or natural science. Hence, the conclusions based on these laws will not be as exact as those based on the law of physics or chemistry. Thus, on the basis of statistical analysis we can talk only in terms of approximation and probability but no in terms of certainty. For example, we can say definitely together it will produce a certain amount of water but we can not conclude any the number 6 in a single throw of an ordinary dice is 1/6, it does not mean that we must get the number 6, 10 times if the same dice is flipped for 60 times. 5 times or 0 times. By this law of statistics we mean that if the experiment of throwing the die is carried on for very large number of times then we can expect to get the number 6 for 1/6th of the time of throws.
(vi) It is likely to be misused. Since statistics deals with figures and figures are innocent and can be easily distorted, manipulated or moulded by the inexpert and unskilled workers or by the motivated, dishonest and unscrupulous persons it is very much likely to be misused in most of the cases. According to W.I. King, “one of the shortcomings of statistics is that they do not bear on their face the label of their quality.” Further, on another occasion Mr. King observes, “Statistics are like clay of which you can make God or a Devil as you please” and “science of statistics is the useful servant but only of great value to those who understand its proper use.”