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Examples of Coefficients and How To Solve Them

By Kathleen Knowles, 01 Oct 2020

You may not think you have a use for coefficients and statistics, but you'd be wrong. Do you know that you do statistics every single day? When you woke up this morning, you made a decision to brush your teeth, to have your bath, and to take breakfast. These things would not have been possible for you without statistics.

Researchers are able to provide information to improve life with the help of their statistical skills. Manufacturing companies use statistics to produce quality products. For example, pharmaceutical companies work hard to produce drugs with the best quality and do sample tests to know how effective drugs are. They're able to do this with the help of statistics.

The government use statistics to determine the winners of elections in countries in the world. The Government collects data during elections and use statistics to know the contestants with the highest votes. Apart from elections, governments are also able to collect useful information over the internet to make important decisions. Without statistics, the invention of the internet wouldn't have been possible.

Coefficients

Data is a collection of information. Bearing this in mind, we can say coefficient is a measure of the characteristic of data.

Types of Coefficients

We can't talk about coefficient without mentioning correlation. In simple words, correlation measures how two variables are related to one another. Two variables might be related to one another positively or negatively.  So you can say two variables are positively related or negatively related.

  • Positive correlation: You compared two variables together and found that as one variable increases, the other variable increases also. It means that the two variables have a positive correlation. We represent a positive correlation between two variables with + 1
  • Negative correlation:   When two variables are compared to one another and it is found that as one variable increases, the other variable decreases. We conclude that the two variables are negatively correlated. This is denoted by -1.

The type of correlation mostly used in the world today is called the Pearson Correlation. This statistic was developed by Karl Pearson and introduced in the 1880s.

Most students and lecturers write it as Pearson Correlation but the full name is Pearson Product Moment Correlation (PPMC). It's a number that measures the strength of the correlation between given variables.

Mathematical Representation

In mathematics, we represent the Pearson correlation with a small letter “r” and a general formula: r = +1/-1. You might be wondering, is it applicable in a real-life scenario? Definitely!

It's widely used all over the world to show the relationship that exists between the two variables. We plot a graph with the values of the two variables and draw a line of best fit to connect all the possible points on our graph.

Our Pearson coefficient “r” value indicates how these points are far away from the line of best fit. The value of “r” determined would always be between +1 and -1 where:

  • r = +1/-1 represents the assumption that all data points are on our line of best fit. This means that if almost all our points lie on our line of best fit, the association between our variables is a very strong one.
  • r = 0 means that there is no association between our variables.
  • If we have the value of r to be between +1 and -1, it means some of our data points are not on the line of best fit. The closer the value of r tends to 0, means there are more data points off our line of best fit.

Variables that can be used with Pearson Correlation

We don’t use this statistic with all types of variables to show relationships. The two types of variables that we can use Pearson to show their association is ones where a variable can be measured in ratio and the other variable measured on an interval.

Let’s say, for example, the age of a person and the blood sugar level. Age is a variable measured on intervals in years while blood sugar level is a variable measured in a ratio in mmol/L concentration.

Where n = 2 and N = our sample case, let’s consider an example where we show the value of “r” from the following table:

We’ll determine the values for each term in the formula above. For using that formula we need to compute Σ(X*Y), Σ(X), Σ(Y), Σ(X²), Σ(Y²). The table below shows the computed values of all the summations mentioned above.

From our table we get:

  • Σ(X) = 46
  • Σ(Y) = 205
  • Σ(X*Y) = 1640
  • Σ(X²) = 766
  • Σ(Y²) = 14297
  • N is the sample size, in our case = 3

= 3(1640) — (46 × 205) / [√[[3(766) — (46²)] × [3(14297) — 205²]]]

= 0.0026

Coefficient “r” is almost 0. As you can see, we can conclude that there is no relationship between the two variables we tested.

Reliability Coefficients

When last did you check your weight on a scale? Did you trust the scale readings? These questions bring us to our next coefficient, “Reliability Coefficient.”

This coefficient measures the accuracy of a measuring instrument after second or more measurements. If the second reading you got from the scale differs largely from the first reading you took a month or two ago, then there is either an error with the scale or something is wrong with your body.

Test-Retest Reliability

We're going to illustrate this concept with an example. Supposed you're a teacher and you held a test for your students 3 months ago. And recently you just conducted another test for them on the same subject. All other things being equal, students that scored "As’’ in the first test should score “As” in the second test and so on The point here is that there should be no obvious change in the students’ performance.

The test-retest reliability coefficient shows the correlation between the first test and the second test or re-test. Since this is also a measure of coefficient, we use the Pearson coefficient statistic to show how the two tests relate to each other.

Inter-rater Reliability Method

This type of coefficient shows us the extent to which two or more raters, like observers or examiners, agree. When we get a high value, it means a high degree of agreement and vice versa. If two raters agree, IRR would be 1, or we say 100% and if they disagree, IRR is 0 or 0%.

There are many statistics for determining this coefficient, the statistic you would use depend on the nature of the variables. A simple statistic that we use for simple data is the Percent Agreement.

Follow the steps below to determine the inter-rater reliability between raters. To better understand these rules let’s work it with an example.

Rules:
  1. Count and write down the number of ratings in the agreement. In our example above, it is 3.
  2. You will need to count the total number of ratings. In our example, that is 5.
  3. You will then divide the total by the number in agreement. In our example above, we get 2/5.
  4. Convert the fraction to a percentage. That is 2/5 = 40%

Conclusion

Statistics is part of life. To stay alive in itself is doing statistics. The information we get through statistics problems keeps us informed about our surrounding world. We live in an age today, where information drives everything, and statistics make getting this information possible.

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