UNIVARIATE ANALYSIS ASSIGNMENT HELP

One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the central value of an average. It is useful to have a single value or an average. It is useful to have a single value which measures the central tendency, both to condense the information contained in a sample and for the purpose of comparison. The value of the measure of central tendency is two fold. First, it is an ‘average’ which represents all of the scores made by the group, and as such gives a concise description of the performance of the group as whole, second, it enables us to compare two or more groups in terms of typical performance. The commonly used central measures are:

• the arithmetic mean
• the median
• the mode
• the geometric mean
• the quartiles and the deciles

The Arithmetic Mean:

To obtain a measure of central tendency which gives equal weight to all values we need, to turn to the arithmetic mean. The term arithmetic mean or simply mean is known as average. The simplest method of finding the arithmetic mean is by dividing the sum of the scores, by the number of items.

• If X1, X2,……………………….. Xn are N scores
• then arithmetic mean = (X1 + X2+…………+Xn) / N
• M = X = (∑x)/ N

Assumed Mean Method:

When the scores are large numbers the calculation become difficult. In that case a number is assumed as mean, generally the assumed mean is taken as the mid point of the interval of maximum frequencies. Deviations of the scores from the assumed mean are taken. These deviations are multiplied by the respective frequency and the following formula is applied.

• X = assumed mean + ∑fx/ N
• Where x is the deviation from the assumed mean. We can also calculate the arithmetic mean by the formula.
• X = assumed mean + ∑fx Xi/ N
• Where x is the mid point of the class and i is the length of class interval. This method is known as Step deviation method.

The Median

The median is a point on a scale such that half the observation fall above it and half below it e,g. the observation 2, 7, 16, 19, 20, 25 and 27 are arranged in order of magnitude. The median is 19, three observations fall above it and three below it.

The Mode

In a simple ungrouped series of measures the ‘crude’ or ‘empirical’ mode is that single measure or score which occurs most frequently.

• If the data is grouped into a frequency distribution the crude mode is usually taken to be the mid point of that interval which has the largest frequency.
• The mode is given by the following formula.

Mode = l + ( f1 – f0)*i /(2f1 – f0 – f2) Where l is lower limit of modal class, f1 is frequency of the modal class, f0 is frequency of the class preceding the modal class, f2 is frequency of the class succeeding the modal class.

Geometric Mean:

Geometric mean is defined as the Nth root of the product of N items of values.

• G.M = N√(X1) x (X2) x ………. (Xn)
• where X1, X2, ………………..Xn etc. refer to the various items of the series.
• log G.M = log [(X1 x X2 ……. x Xn)] = ∑logX/N
• GM = Antilog [ ∑logX ]/Nin continuous series

where X is the mid point of the continuous series.

Quartiles and Deciles:

If the data is arranged in ascending order than this organized data is typically termed as scale of measurement. On this scale a point below which a specified proportion of cases fall is known as a specified of cases fall is known as a measures of relative position. All the quartiles, deciles etc. are the different measure of relative position.

• Quartiles: The quartile divide the distribution into four equal parts only three point are required.
• Deciles: Deciles divide the distribution into ten equal parts D1, D2…………. D9 denote the first decile, second decile,…….. ninth decile. To divide a distance into ten, equals only nine point will be required.

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