# A range of numbers is used everyday in describing the relationship

### eWorkshop - Online Teaching Resource

Number sense involves an understanding of the many relationships that Students must learn how numbers are interconnected and how numbers can be used. Here is how they differ and how they are used in daily life. for your vacation destination, are all numbers that are important in everyday life. The word correlation is used in everyday life to denote some form of association. If a curved line is needed to express the relationship, other and more complicated . If we wish to label the strength of the association, for absolute values of r, Correlation describes the strength of an association between two variables, and.

It tells you what most of the pieces of data are doing within a set of information.

Mean Is Average The mean is commonly referred to as average, but it is not the only kind of average. The mean is often used in research, academics and in sports.

When you watch a baseball game and you see the player's batting average, that number represents the total number of hits divided by the number of times at bat. In other words, that number is the mean. In school, the final grade you get in a course is usually a mean.

This mean represents the total number of points you scored in the class divided by the number of possible points.

This is the classic type of average — when your overall performance on many items is evaluated with a single number. One is to see whether two measurement variables are associated with each other; whether as one variable increases, the other tends to increase or decrease.

You summarize this test of association with the P value. In some cases, this addresses a biological question about cause-and-effect relationships; a significant association means that different values of the independent variable cause different values of the dependent.

An example would be giving people different amounts of a drug and measuring their blood pressure. The null hypothesis would be that there was no relationship between the amount of drug and the blood pressure.

If you reject the null hypothesis, you would conclude that the amount of drug causes the changes in blood pressure. In this kind of experiment, you determine the values of the independent variable; for example, you decide what dose of the drug each person gets.

The exercise and pulse data are an example of this, as I determined the speed on the elliptical machine, then measured the effect on pulse rate. In other cases, you want to know whether two variables are associated, without necessarily inferring a cause-and-effect relationship. In this case, you don't determine either variable ahead of time; both are naturally variable and you measure both of them.

If you find an association, you infer that variation in X may cause variation in Y, or variation in Y may cause variation in X, or variation in some other factor may affect both Y and X. An example would be measuring the amount of a particular protein on the surface of some cells and the pH of the cytoplasm of those cells. If the protein amount and pH are correlated, it may be that the amount of protein affects the internal pH; or the internal pH affects the amount of protein; or some other factor, such as oxygen concentration, affects both protein concentration and pH.

Often, a significant correlation suggests further experiments to test for a cause and effect relationship; if protein concentration and pH were correlated, you might want to manipulate protein concentration and see what happens to pH, or manipulate pH and measure protein, or manipulate oxygen and see what happens to both. The amphipod data are another example of this; it could be that being bigger causes amphipods to have more eggs, or that having more eggs makes the mothers bigger maybe they eat more when they're carrying more eggs?

### Correlation and linear regression - Handbook of Biological Statistics

The second goal of correlation and regression is estimating the strength of the relationship between two variables; in other words, how close the points on the graph are to the regression line. You summarize this with the r2 value.

You would also want to know whether there's a tight relationship high r2which would tell you that air temperature is the main factor affecting running speed; if the r2 is low, it would tell you that other factors besides air temperature are also important, and you might want to do more experiments to look for them. You might also want to know how the r2 for Agama savignyi compared to that for other lizard species, or for Agama savignyi under different conditions.

The third goal of correlation and regression is finding the equation of a line that fits the cloud of points. You can then use this equation for prediction. For example, if you have given volunteers diets with to mg of salt per day, and then measured their blood pressure, you could use the regression line to estimate how much a person's blood pressure would go down if they ate mg less salt per day.

Correlation versus linear regression The statistical tools used for hypothesis testing, describing the closeness of the association, and drawing a line through the points, are correlation and linear regression. Unfortunately, I find the descriptions of correlation and regression in most textbooks to be unnecessarily confusing. Some statistics textbooks have correlation and linear regression in separate chapters, and make it seem as if it is always important to pick one technique or the other.

I think this overemphasizes the differences between them. Other books muddle correlation and regression together without really explaining what the difference is. There are real differences between correlation and linear regression, but fortunately, they usually don't matter. Correlation and linear regression give the exact same P value for the hypothesis test, and for most biological experiments, that's the only really important result.

## Australian Bureau of Statistics

So if you're mainly interested in the P value, you don't need to worry about the difference between correlation and regression. Be aware that my approach is probably different from what you'll see elsewhere. The main difference between correlation and regression is that in correlation, you sample both measurement variables randomly from a population, while in regression you choose the values of the independent X variable.

For example, let's say you're a forensic anthropologist, interested in the relationship between foot length and body height in humans. If you find a severed foot at a crime scene, you'd like to be able to estimate the height of the person it was severed from. You measure the foot length and body height of a random sample of humans, get a significant P value, and calculate r2 to be 0.

This is a correlation, because you took measurements of both variables on a random sample of people. The r2 is therefore a meaningful estimate of the strength of the association between foot length and body height in humans, and you can compare it to other r2 values.

## Handbook of Biological Statistics

You might want to see if the r2 for feet and height is larger or smaller than the r2 for hands and height, for example. As an example of regression, let's say you've decided forensic anthropology is too disgusting, so now you're interested in the effect of air temperature on running speed in lizards. This is a regression, because you decided which temperatures to use. You'll probably still want to calculate r2, just because high values are more impressive.

But it's not a very meaningful estimate of anything about lizards.

This is because the r2 depends on the values of the independent variable that you chose. For the exact same relationship between temperature and running speed, a narrower range of temperatures would give a smaller r2. Here are three graphs showing some simulated data, with the same scatter standard deviation of Y values at each value of X.

As you can see, with a narrower range of X values, the r2 gets smaller. If you did another experiment on humidity and running speed in your lizards and got a lower r2, you couldn't say that running speed is more strongly associated with temperature than with humidity; if you had chosen a narrower range of temperatures and a broader range of humidities, humidity might have had a larger r2 than temperature.

Simulated data showing the effect of the range of X values on the r2. For the exact same data, measuring Y over a smaller range of X values yields a smaller r2.

If you try to classify every experiment as either regression or correlation, you'll quickly find that there are many experiments that don't clearly fall into one category. For example, let's say that you study air temperature and running speed in lizards. You go out to the desert every Saturday for the eight months of the year that your lizards are active, measure the air temperature, then chase lizards and measure their speed. You haven't deliberately chosen the air temperature, just taken a sample of the natural variation in air temperature, so is it a correlation?

But you didn't take a sample of the entire year, just those eight months, and you didn't pick days at random, just Saturdays, so is it a regression? In doing so, students find deeper meaning in a text and have the opportunity to present their ideas through oral and written responses. Today's junior students are comfortable with technology and use a wide range of electronic and online communication devices to send and receive information. Teachers must purposefully integrate technology into all aspects of classroom teaching and learning to help junior students critically assess, filter, and interpret the information they are receiving and develop the skills they need to use technology confidently and appropriately in the classroom and beyond.

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