# Rank Correlation Coefficient Problem And Solution PDF

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## Spearman Rank Correlation Formulas For Repeated Ranks PDF Free Download

### Rank Correlation Definition

For the calculation of product moment correlation coefficient characters must be measurable. In many practical situations, characters are not measurable.

They are qualitative characteristics and individuals or items can be ranked in order of their merits. This type of situation occurs when we deal with the qualitative study such as honesty, beauty, voice, etc.

For example, contestants of a singing competition may be ranked by judge according to their performance.

In another example, students may be ranked in different subjects according to their performance in tests.

Arrangement of individuals or items in order of merit or proficiency in the possession of a certain characteristic is called ranking and the number indicating the position of individuals or items is known as rank.

If ranks of individuals or items are available for two characteristics then correlation between ranks of these two characteristics is known as rank correlation. With the help of rank correlation, we find the association between two qualitative characteristics.

As we know that the Karl Pearson’s correlation coefficient gives the intensity of linear relationship between two variables and Spearman’s rank correlation coefficient gives the concentration of association between two qualitative characteristics.

In fact Spearman’s rank correlation coefficient measures the strength of association between two ranked variables. Derivation of the Spearman’s rank correlation coefficient formula is discussed in the following section.

#### Merits and Demerits of Rank Correlation Coefficient

Merits of Rank Correlation Coefficient:

1. Spearman’s rank correlation coefficient can be interpreted in the same way as the Karl Pearson’s correlation coefficient;

2. It is easy to understand and easy to calculate;

3. If we want to see the association between qualitative characteristics, rank correlation coefficient is the only formula;

4. Rank correlation coefficient is the non-parametric version of the Karl Pearson’s product moment correlation coefficient; and

5. It does not require the assumption of the normality of the population from which the sample observations are taken.

Demerits of Rank Correlation Coefficient

1. Product moment correlation coefficient can be calculated for bivariate frequency distribution but rank correlation coefficient cannot be calculated; and

2. If n >30, this formula is time consuming.

#### Tied Or Repeated Ranks In Section

it was assumed that two or more individuals or units do not have same rank. But there might be a situation when two or more individuals have same rank in one or both characteristics, then this situation is said to be tied.

If two or more individuals have same value, in this case common ranks are assigned to the repeated items. This common rank is the average of ranks they would have received if there were no repetition.

For example we have a series 50, 70, 80, 80, 85, 90 then 1st rank is assigned to 90 because it is the biggest value then 2nd to 85, now there is a repetition of 80 twice.

Since both values are same so the same rank will be assigned which would be average of the ranks that we would have assigned if there were no repetition. Thus, both 80 will receive the average of 3 and 4 i.e

### Spearman’s Correlation

#### Monotonic function

To understand Spearman’s correlation it is necessary to know what a monotonic function is. A monotonic function is one that either never increases or never decreases as its independent variable increases. The following graphs illustrate monotonically functions:

Monotonically increasing Monotonically decreasing Not monotonic(rx And ry)

• Monotonically increasing – as the x variable increases the y variable never decreases;
• Monotonically decreasing – as the x variable increases the y variable never increases;
• Not monotonic – as the x variable increases the y variable sometimes decreases and sometimes increases

#### Spearman’s correlation coefficient

Spearman’s correlation coefficient is a statistical measure of the strength of a monotonic relationship between paired data. In a sample it is denoted by and is by design constrained as follows

And its interpretation is similar to that of Pearsons, e.g. the closer is to the stronger the monotonic relationship. Correlation is an effect size and so we can verbally describe the strength of the correlation using the following guide for the absolute value of:

.00-.19 “very weak”
.20-.39 “weak”
.40-.59 “moderate”
.60-.79 “strong”
.80-1.0 “very strong”

The calculation of Spearman’s correlation coefficient and subsequent significance testing of it requires the following data assumptions to hold:

interval or ratio level or ordinal;
monotonically related.

Note, unlike Pearson’s correlation, there is no requirement of normality and hence it is a nonparametric statistic.

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