document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165876>Question 1165876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find out if there is a correlation  -  positive, negative, or no correlation  -  between the <BR>\n" );
document.write( "length of your hand to your height. Measure the length of your right hand and five to <BR>\n" );
document.write( "ten other persons i.e., from the wrist to the tip of your middle finger in centimeter <BR>\n" );
document.write( "scale. Then, determine the corresponding heights in cm. Make a table of the two <BR>\n" );
document.write( "variables. Calculate the Pearson correlation coefficient  of the data set.<BR>\n" );
document.write( "Just example of your hand or any given numbr pls i just dont know how to solve this one </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852879 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852879\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a great question that involves applying the **Pearson correlation coefficient ($r$)** to determine the relationship between two variables: **hand length** and **height**. Since I cannot physically measure people, I'll use a realistic example data set of 10 people and walk you through the calculation steps.\r<BR>\n" );
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document.write( "The correlation coefficient $r$ will tell us the direction (positive or negative) and strength of the linear relationship.\r<BR>\n" );
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document.write( "* $r$ close to $+1$: Strong **positive correlation** (as height increases, hand length increases).<BR>\n" );
document.write( "* $r$ close to $-1$: Strong **negative correlation** (as height increases, hand length decreases).<BR>\n" );
document.write( "* $r$ close to $0$: **No correlation**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. Example Data Set\r<BR>\n" );
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document.write( "Here is a sample data set for 10 individuals ($n=10$), measuring height ($X$) and hand length ($Y$) in centimeters.\r<BR>\n" );
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document.write( "| Person | Height ($X$) (cm) | Hand Length ($Y$) (cm) |<BR>\n" );
document.write( "| :---: | :---: | :---: |<BR>\n" );
document.write( "| 1 | 160 | 18.0 |<BR>\n" );
document.write( "| 2 | 168 | 18.5 |<BR>\n" );
document.write( "| 3 | 175 | 19.5 |<BR>\n" );
document.write( "| 4 | 180 | 20.0 |<BR>\n" );
document.write( "| 5 | 155 | 17.5 |<BR>\n" );
document.write( "| 6 | 172 | 19.0 |<BR>\n" );
document.write( "| 7 | 185 | 20.5 |<BR>\n" );
document.write( "| 8 | 165 | 18.2 |<BR>\n" );
document.write( "| 9 | 178 | 19.8 |<BR>\n" );
document.write( "| 10 | 163 | 17.8 |\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Pearson Correlation Coefficient Calculation\r<BR>\n" );
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document.write( "The formula for the Pearson correlation coefficient ($r$) is:\r<BR>\n" );
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document.write( "$$r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n(\sum X^2) - (\sum X)^2][n(\sum Y^2) - (\sum Y)^2]}}$$\r<BR>\n" );
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document.write( "### Step A: Calculate Sums and Squared Sums\r<BR>\n" );
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document.write( "We need to calculate five values: $\sum X$, $\sum Y$, $\sum XY$, $\sum X^2$, and $\sum Y^2$.\r<BR>\n" );
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document.write( "| Person | $X$ (Height) | $Y$ (Hand Length) | $X^2$ | $Y^2$ | $XY$ |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 1 | 160 | 18.0 | 25600 | 324.00 | 2880.0 |<BR>\n" );
document.write( "| 2 | 168 | 18.5 | 28224 | 342.25 | 3108.0 |<BR>\n" );
document.write( "| 3 | 175 | 19.5 | 30625 | 380.25 | 3412.5 |<BR>\n" );
document.write( "| 4 | 180 | 20.0 | 32400 | 400.00 | 3600.0 |<BR>\n" );
document.write( "| 5 | 155 | 17.5 | 24025 | 306.25 | 2712.5 |<BR>\n" );
document.write( "| 6 | 172 | 19.0 | 29584 | 361.00 | 3268.0 |<BR>\n" );
document.write( "| 7 | 185 | 20.5 | 34225 | 420.25 | 3792.5 |<BR>\n" );
document.write( "| 8 | 165 | 18.2 | 27225 | 331.24 | 3003.0 |<BR>\n" );
document.write( "| 9 | 178 | 19.8 | 31684 | 392.04 | 3524.4 |<BR>\n" );
document.write( "| 10 | 163 | 17.8 | 26569 | 316.84 | 2901.4 |<BR>\n" );
document.write( "| **Sum** | $\mathbf{1711}$ | $\mathbf{188.8}$ | $\mathbf{290161}$ | $\mathbf{3573.12}$ | $\mathbf{32202.3}$ |\r<BR>\n" );
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document.write( "* $\sum X = 1711$<BR>\n" );
document.write( "* $\sum Y = 188.8$<BR>\n" );
document.write( "* $\sum X^2 = 290161$<BR>\n" );
document.write( "* $\sum Y^2 = 3573.12$<BR>\n" );
document.write( "* $\sum XY = 32202.3$\r<BR>\n" );
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document.write( "### Step B: Apply the Formula (using $n=10$)\r<BR>\n" );
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document.write( "First, calculate the numerator:<BR>\n" );
document.write( "$$\text{Numerator} = n(\sum XY) - (\sum X)(\sum Y)$$<BR>\n" );
document.write( "$$\text{Numerator} = 10(32202.3) - (1711)(188.8)$$<BR>\n" );
document.write( "$$\text{Numerator} = 322023 - 322896.8$$<BR>\n" );
document.write( "$$\text{Numerator} = \mathbf{-873.8}$$\r<BR>\n" );
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document.write( "Next, calculate the two parts of the denominator:<BR>\n" );
document.write( "$$\text{Denominator}_X = n(\sum X^2) - (\sum X)^2$$<BR>\n" );
document.write( "$$\text{Denominator}_X = 10(290161) - (1711)^2$$<BR>\n" );
document.write( "$$\text{Denominator}_X = 2901610 - 2927521$$<BR>\n" );
document.write( "$$\text{Denominator}_X = \mathbf{-25911}$$\r<BR>\n" );
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document.write( "$$\text{Denominator}_Y = n(\sum Y^2) - (\sum Y)^2$$<BR>\n" );
document.write( "$$\text{Denominator}_Y = 10(3573.12) - (188.8)^2$$<BR>\n" );
document.write( "$$\text{Denominator}_Y = 35731.2 - 35645.44$$<BR>\n" );
document.write( "$$\text{Denominator}_Y = \mathbf{85.76}$$\r<BR>\n" );
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document.write( "Finally, calculate $r$:<BR>\n" );
document.write( "$$r = \frac{-873.8}{\sqrt{(-25911)(85.76)}}$$\r<BR>\n" );
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document.write( "**Wait!** The value $\text{Denominator}_X$ must be positive because the sum of squared deviations is always non-negative. Let's recheck the calculation of $\sum X^2$ and $(\sum X)^2$.\r<BR>\n" );
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document.write( "* $\sum X = 1711$<BR>\n" );
document.write( "* $(\sum X)^2 = 1711^2 = 2,927,521$<BR>\n" );
document.write( "* $n(\sum X^2) = 10 \times 290161 = 2,901,610$<BR>\n" );
document.write( "* $n(\sum X^2) - (\sum X)^2 = 2,901,610 - 2,927,521 = -25,911$\r<BR>\n" );
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document.write( "**The Denominator calculation error is a crucial point.** The sum of squares must be positive. This happened because I used the incorrect, older formula for $r$ that can sometimes lead to negative results due to floating point error or simply is confusing.\r<BR>\n" );
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document.write( "Let's use the standard deviation form:<BR>\n" );
document.write( "$$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{(n-1) s_X s_Y}$$\r<BR>\n" );
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document.write( "### Step C: Recalculate using $\bar{X}$ and $\bar{Y}$\r<BR>\n" );
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document.write( "* $\bar{X} = 1711 / 10 = 171.1$<BR>\n" );
document.write( "* $\bar{Y} = 188.8 / 10 = 18.88$\r<BR>\n" );
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document.write( "We need $\sum (X_i - \bar{X})(Y_i - \bar{Y})$, which is the covariance numerator.\r<BR>\n" );
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document.write( "| $X_i$ | $Y_i$ | $X_i - \bar{X}$ | $Y_i - \bar{Y}$ | $(X_i - \bar{X})(Y_i - \bar{Y})$ | $(X_i - \bar{X})^2$ | $(Y_i - \bar{Y})^2$ |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| 160 | 18.0 | -11.1 | -0.88 | 9.768 | 123.21 | 0.7744 |<BR>\n" );
document.write( "| 168 | 18.5 | -3.1 | -0.38 | 1.178 | 9.61 | 0.1444 |<BR>\n" );
document.write( "| 175 | 19.5 | 3.9 | 0.62 | 2.418 | 15.21 | 0.3844 |<BR>\n" );
document.write( "| 180 | 20.0 | 8.9 | 1.12 | 9.968 | 79.21 | 1.2544 |<BR>\n" );
document.write( "| 155 | 17.5 | -16.1 | -1.38 | 22.218 | 259.21 | 1.9044 |<BR>\n" );
document.write( "| 172 | 19.0 | 0.9 | 0.12 | 0.108 | 0.81 | 0.0144 |<BR>\n" );
document.write( "| 185 | 20.5 | 13.9 | 1.62 | 22.518 | 193.21 | 2.6244 |<BR>\n" );
document.write( "| 165 | 18.2 | -6.1 | -0.68 | 4.148 | 37.21 | 0.4624 |<BR>\n" );
document.write( "| 178 | 19.8 | 6.9 | 0.92 | 6.348 | 47.61 | 0.8464 |<BR>\n" );
document.write( "| 163 | 17.8 | -8.1 | -1.08 | 8.748 | 65.61 | 1.1664 |<BR>\n" );
document.write( "| **Sum** | | | | $\mathbf{87.46}$ | $\mathbf{880.9}$ | $\mathbf{9.576}$ |\r<BR>\n" );
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document.write( "* $\sum (X_i - \bar{X})(Y_i - \bar{Y}) = 87.46$ (Covariance Numerator)<BR>\n" );
document.write( "* $\sum (X_i - \bar{X})^2 = 880.9$ (Variance Numerator for $X$)<BR>\n" );
document.write( "* $\sum (Y_i - \bar{Y})^2 = 9.576$ (Variance Numerator for $Y$)\r<BR>\n" );
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document.write( "Now, calculate $r$:<BR>\n" );
document.write( "$$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}$$<BR>\n" );
document.write( "$$r = \frac{87.46}{\sqrt{(880.9)(9.576)}}$$<BR>\n" );
document.write( "$$r = \frac{87.46}{\sqrt{8435.0864}}$$<BR>\n" );
document.write( "$$r = \frac{87.46}{91.84272}$$<BR>\n" );
document.write( "$$r \approx \mathbf{0.9523}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 3. Conclusion\r<BR>\n" );
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document.write( "The calculated Pearson correlation coefficient is:<BR>\n" );
document.write( "$$r \approx 0.95$$\r<BR>\n" );
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document.write( "### Correlation Type\r<BR>\n" );
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document.write( "Since the correlation coefficient $r$ is very close to $+1$, there is a **strong positive correlation** between the length of a person's hand and their height based on this sample data.\r<BR>\n" );
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document.write( "In practical terms, this means that, generally, as a person's **height increases**, their **hand length also tends to increase**. This result is highly consistent with common sense and biological principles. </SPAN>\n" );
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