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Assignment 2: List the advantages and disadvantages of Measures of Central Tendency vis a vis Measures of Dispersion. When would you use either? Give a brief and precise report on this issue Measures of central tendency A measure of central tendency is a summary statistic that denotes the center point or typical value of a dataset. Measures of central tendency indicate where most values in a distribution fall. Measures of central tendency include the mean, mode and median but are applicable under different conditions as will be discussed subsequently. Mean The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. x X´ = 1

+ x2 …+x n n

Advantages of the mean 1. It is rigidly defined. 2. It is easy to calculate and simple to follow. 3. It is based on all the observations. 4. It is determined for almost every kind of data. 5. It is finite and not indefinite. 6. It is readily put to algebraic treatment. 7. It is least affected by fluctuations of sampling. Disadvantages of the mean 1. The arithmetic mean is highly affected by extreme values. 2. It cannot average the ratios and percentages properly. 3. It is not an appropriate average for highly skewed distributions. 4. It cannot be computed accurately if any item is missing. 5. The mean sometimes does not coincide with any of the observed value. When not to use the mean When the data has several outliers, it is not advisable to use the mean as it is susceptible to these outliers. Also, when the data has so many repeated values, the mean is also no applicable.

Median Median is the value which occupies the middle position when all the observations are arranged in an ascending/descending order. It divides the frequency distribution exactly into two halves. Advantages of median 1. It is easy to compute and comprehend. 2. It is not distorted by outliers/skewed data. 3. It can be determined for ratio, interval, and ordinal scale. Disadvantages of median 1. It does not take into account the precise value of each observation and hence does not use all information available in the data. 2. Unlike mean, median is not amenable to further mathematical calculation and hence is not used in many statistical tests. 3. If we pool the observations of two groups, median of the pooled group cannot be expressed in terms of the individual medians of the pooled groups. When to use the median Median is preferred to mean when 1. There are few extreme scores in the distribution. 2. Some scores have undetermined values. 3. There is an open ended distribution. 4. Data are measured in an ordinal scale. Mode The mode is the most frequent score in our data set. Advantages of the mode 1. It is the only measure of central tendency that can be used for data measured in a nominal scale.

2. It can be calculated easily. Disadvantages of mode 1. It is not used in statistical analysis as it is not algebraically defined and the fluctuation in the frequency of observation is more when the sample size is small. 2. It is not unique, so it leaves us with problems when we have two or more values that share the highest frequency When to use the mode The mode is the only measure of central tendency that you can use with categorical data—such as the most preferred color of clothing. Also, mode is the preferred measure when data are measured in a nominal scale. Measures of dispersion The measures of dispersion show the scatterings of the data. Measures of dispersion tell the variation of the data from one another and gives a clear idea about the distribution of the data. Common measures of dispersion include: i)

Range

ii)

Quartile deviation

iii)

Standard deviation

iv)

Mean deviation

Range Range is the difference between two extreme observations of the data set. If X max and Xmin are the two extreme observations then Range = X max – X min Advantages of Range 1. It is the simplest of the measure of dispersion 2. Easy to calculate 3. Easy to understand

4. Independent of change of origin Disadvantages of Range 

It is based on two extreme observations. Hence, get affected by fluctuations



A range is not a reliable measure of dispersion



Dependent on change of scale

Quartile Deviation The quartiles divide a data set into quarters. The first quartile, (Q 1) is the middle number between the smallest number and the median of the data. The second quartile, (Q 2) is the median of the data set. The third quartile, (Q3) is the middle number between the median and the largest number. Quartile deviation or semi-inter-quartile deviation is Q = ½ × (Q3 – Q1) Advantages of Quartile Deviation 1. All the drawbacks of Range are overcome by quartile deviation 2. It uses half of the data 3. Independent of change of origin 4. The best measure of dispersion for open-end classification Disadvantages of Quartile Deviation 1. It ignores 50% of the data 2. Dependent on change of scale 3. Not a reliable measure of dispersion Standard Deviation A standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. It is denoted by a Greek letter sigma, σ. It is also referred to as root mean square deviation. The standard deviation is given as

√∑ n

σ=

i=1

f i ( y i− ´y ) 2 n−1

NOTE: The square of the standard deviation is the variance. It is also a measure of dispersion. Advantages of Standard Deviation 1. Squaring the deviations overcomes the drawback of ignoring signs in mean deviations

2. Suitable for further mathematical treatment 3. Least affected by the fluctuation of the observations 4. The standard deviation is zero if all the observations are constant 5. Independent of change of origin Disadvantages of Standard Deviation 1. Not easy to calculate 2. Difficult to understand for a layman 3. Dependent on the change of scale Mean Deviation Mean deviation is the arithmetic mean of the absolute deviations of the observations from a measure of central tendency. If x1, x2, … , xn are the set of observation, then the mean deviation of x about the average A (mean, median, or mode) is Mean deviation from average A =

1 n

[

n

]

f i| xi − A| ∑ i=1

Merits of Mean Deviation 1. Based on all observations 2. It provides a minimum value when the deviations are taken from the median 3. Independent of change of origin Demerits of Mean Deviation 1. Not easily understandable 2. Its calculation is not easy and time-consuming 3. Dependent on the change of scale 4. Ignorance of negative sign creates artificiality and becomes useless for further mathematical treatment References Anilkumar, D. R. P., & NV, S. (2013). Refining Measure of Central Tendency and Dispersion. IOSR Journal Of Mathematics, 6(1), 1-4.

Deshpande, S., Gogtay, N. J., & Thatte, U. M. (2016). Measures of central tendency and dispersion. Journal of the Association of Physicians of India, 64, 64-66. Manikandan,

S.

(2011).

Measures

Pharmacotherapeutics, 2(4), 315.

of

dispersion. Journal

of

Pharmacology

and...