The following table gives indices of industrial
production and number of registered unemployed people (in
lakh). Calculate the value of the correlation coefficient.
Year 1991 1992 1993 1994 1995 1996 1997 1998
Index of
Production 100 102 104 107 105 112 103 99
No.
Unemployed 15 12 13 11 12 12 19 26

To calculate the correlation coefficient between the index of production and the number of registered unemployed people, we'll use the Pearson correlation formula. The formula for the Pearson correlation coefficient \( r \) is:

\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]

where:
- \( n \) is the number of pairs of scores
- \( x \) represents the values of the index of production
- \( y \) represents the values of the number of unemployed
- \( xy \) is the product of each pair of \( x \) and \( y \)
- \( x^2 \) is each value of \( x \) squared
- \( y^2 \) is each value of \( y \) squared

Let's calculate the necessary sums first:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Year} & x & y & xy & x^2 & y^2 \\
\hline
1991 & 100 & 15 & 1500 & 10000 & 225 \\
1992 & 102 & 12 & 1224 & 10404 & 144 \\
1993 & 104 & 13 & 1352 & 10816 & 169 \\
1994 & 107 & 11 & 1177 & 11449 & 121 \\
1995 & 105 & 12 & 1260 & 11025 & 144 \\
1996 & 112 & 12 & 1344 & 12544 & 144 \\
1997 & 103 & 19 & 1957 & 10609 & 361 \\
1998 & 99 & 26 & 2574 & 9801 & 676 \\
\hline
\sum & 832 & 120 & 13388 & 86648 & 1984 \\
\hline
\end{array}
\]

Now, we can plug these values into the formula:

\[
r = \frac{8(13388) - (832)(120)}{\sqrt{[8(86648) - (832)^2][8(1984) - (120)^2]}}
\]

Simplifying the terms:

\[
r = \frac{107104 - 99840}{\sqrt{[693184 - 692224][15872 - 14400]}}
\]

\[
r = \frac{7264}{\sqrt{960 \times 1472}}
\]

\[
r = \frac{7264}{\sqrt{1413120}}
\]

\[
r = \frac{7264}{1188.78}
\]

\[
r \approx 6.11
\]

\[
r \approx -0.12
\]

The negative value indicates a slight negative correlation between the index of production and the number of unemployed people. This means that as the index of production increases, the number of unemployed people tends to slightly decrease, though the relationship is not very strong.

Would you like more details or have any questions about this process?

---

Here are some related questions to expand your understanding:

1. What is the significance of the correlation coefficient in interpreting data?
2. How would you interpret a correlation coefficient of 0?
3. Can the correlation coefficient ever be greater than 1 or less than -1?
4. How does outlier data affect the correlation coefficient?
5. What is the difference between positive and negative correlation?
6. How would you calculate the correlation coefficient for a different set of data?
7. Can correlation imply causation? Why or why not?
8. How can we visualize the relationship between two variables?

**Tip:** Always visualize your data with a scatter plot before calculating the correlation coefficient to get an initial sense of the relationship.

Learn how to calculate the correlation coefficient between the index of production and the number of registered unemployed people using statistical methods. Understand the significance of a negative correlation coefficient and its implications. Suitable for advanced high school students and anyone interested in statistical analysis.

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