Bivariate data Notes

Bivariate Data Describes Relationships Between Two Variables

1. In many real situations you measure two numerical variables for each individual or item.
2. For instance, you might record a student's music exam score and the same student's art exam score, or the number of laps a student can run and the hours of TV they watch each week.

1. Usually, one variable is treated as the independent variable (the input) and the other as the dependent variable (the output).
2. On graphs, the independent variable is plotted on the $x$-axis and the dependent variable is plotted on the $y$-axis.

Scatter Diagrams Reveal Patterns And Correlation

1. The main graph used for bivariate data is a scatter diagram (or scatter plot).
2. Each pair $(x, y)$ is plotted as a single point.
3. A scatter diagram helps you judge:
   1. the direction of the relationship (upward or downward trend),
   2. the strength (how tightly points cluster around a line),
   3. any outliers (points that do not fit the overall pattern).

Positive, Negative, And No Correlation

1. Positive correlation: as $x$ increases, $y$ tends to increase (points slope upward).
2. Negative correlation: as $x$ increases, $y$ tends to decrease (points slope downward).
3. No correlation: there is no clear trend (points look scattered with no direction).

The Mean Point Is A Key Reference: $(\bar{x}, \bar{y})$

1. For a set of bivariate data, calculate the mean of the $x$-values, called $\bar{x}$, and the mean of the $y$-values, called $\bar{y}$.
2. The point $M(\bar{x}, \bar{y})$ is called the mean point.
3. A useful fact is that a sensible line of best fit passes through the mean point $ (\bar{x}, \bar{y})$.
4. This is why many "by eye" methods start by plotting the mean point.

Drawing A Line Of Best Fit By Eye

A common method is:

1. Calculate $\bar{x}$ and $\bar{y}$.
2. Plot $M(\bar{x}, \bar{y})$ on the scatter diagram.
3. Draw a straight line through $M$ that goes through the "middle" of the data so that points are reasonably balanced above and below the line.

Outliers Can Distort Interpretation

1. In bivariate data, you can spot outliers by looking for points that are a large distance from the line of best fit.
2. Outliers can happen due to measurement error, unusual circumstances, or because the relationship is not truly linear.

Example

Laps Run Versus TV Hours Shows Negative Correlation

Data from 7 students:

• $x$: number of laps a student could run: $2, 15, 17, 3, 20, 3, 6$
• $y$: TV hours per week: $13, 7, 5, 12, 4, 13, 11$

A scatter diagram for this data shows a downward trend, meaning negative correlation (students who run more laps tend to watch fewer hours of TV).

The drawn line here is a computed straight-line fit, but the key interpretation is the same as a by-eye line: the overall trend is decreasing, so the correlation is negative.

Covariance Measures The Direction Of A Linear Relationship

1. Correlation can be described visually, but there are also numerical measures.
2. One important measure related to correlation is covariance.
3. To understand covariance, it helps to recall variance and standard deviation for one variable: $$\sigma^{2}=\frac{\sum (x-\bar{x})^{2}}{n}$$ $$\sigma=\sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}$$
4. Variance measures how spread out one-variable data is from the mean.
5. Covariance extends this idea to two variables.

A common formula is: $$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n}$$

Why The Sign Of Covariance Matters

1. Consider the deviations from the mean point $(\bar{x},\bar{y})$.
2. Draw a vertical line through $\bar{x}$ and a horizontal line through $\bar{y}$, creating four quadrants:
   1. TRQ (top-right): $x>\bar{x}$ and $y>\bar{y}$, so $(x-\bar{x})>0$ and $(y-\bar{y})>0$, product is positive.
   2. BLQ (bottom-left): $x<\bar{x}$ and $y<\bar{y}$, so both deviations are negative, product is positive.
   3. TLQ (top-left): one deviation negative, the other positive, product is negative.
   4. BRQ (bottom-right): one deviation positive, the other negative, product is negative.
3. As a result:
   1. if most points lie in TRQ and BLQ, the sum of products tends to be positive, so $s_{xy}>0$ (positive slope trend).
   2. if most points lie in TLQ and BRQ, the sum of products tends to be negative, so $s_{xy}<0$ (negative slope trend).

Using Bivariate Data For Prediction Requires Interpolation, Not Extrapolation

1. A line of best fit is often used to predict values.
2. For example, if you have a relationship between a music score ($x$) and an art score ($y$), you might use the line to estimate an art score for a given music score.
3. However, predictions are only reliable when you predict within the range of the observed data.