Module 1:4 - Quantifying Locations

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A - Introduction[edit]

The location of scores in distributions is significant in understanding distributions. Analyzing data by only calculating measures of central tendencies (the mean, median, and mode) may not produce accurate conclusions without considering score location.

The following outline focuses on the importance of utilizing computational methods to determine score location. It will be further explained using JMP to gain better conceptual understanding.

B - Location of Scores in a Distribution with JMP[edit]

Comparing histogram distribution of Quiz1 and Quiz2, from Dr. Parris' slides in Seeing the location of scores in a distribution with JMP

1 - Test Scores Example[edit]

First, we will look at the distributions of Quiz 1 and Quiz 2 from the Introductory example, using JMP. We will use this information to compare the results of Mary and John's quiz scores by examining measurements such as the mean. Doing this helps put the scores of the different quiz variables into context and understanding the limitations of measurements.

Quiz Distribution[edit]

After selecting Analyze>Distribution>Y,Columns, the resulting histograms indicate

1) On Quiz 1 Mary, with a score of 78, and John, with a score of 83, scored pretty close to each other as well as pretty close to the center of the distribution. They performed about equally.

2) On Quiz 2 Mary, with a score of 89, and John, with a score of 84, scored on both ends of the spectrum, 84 being the lowest score and 89 being 1 score from the top. Mary performed very well and John performed poorly.

However, if each individual is examined by comparing their scores, John did about the same, scoring 83 on Quiz 1 and 84 on Quiz 2. Mary has an exceptionally large difference from her 78 in Quiz 1 to her 89 in Quiz 2. These values are regarded in terms of where they fall along a scale of 100 without considering the scores in relation to other scores for each quiz.

An example of inputting a formula, slides from Dr. Parris' Seeing the location of scores in a distribution with JMP

2 - Function of the Mean in JMP[edit]

There are two ways Dr. Julian Parris discussed to calculate the mean in JMP. Next to the quiz columns, make a new column for averages.

Mary and John achieved the same average overall, as seen in Dr. Parris' video Seeing the location of scores in a distribution with JMP

Creating the Average Function[edit]

1) Right click the new column and under formula, input "[Quiz1 + Quiz2]/2."

  • Remember to also click the outer red box so that the sum of Quiz 1 and Quiz 2 can be properly divided.

IncorrectAvgFormula.png CorrectAvgFormula.png

Using the Mean Function[edit]

1) Under the new column, right click>formula>statistical>mean.

  • Use a comma to enter more than one column to compute the mean.

3 - Beyond the Mean[edit]

The raw scores,used for averaging overall performance across Quizzes, know nothing about the distribution of scores within separate Quiz variables. Therefore the average of raw scores also knows nothing about relative location of a score within a distribution.

The mean of a given distribution is a descriptive calculation, but there are times when the mean of a distribution conveys no significantly useful information because it only indicates average without relativity. Thus, we need another method of numerically interpreting overall performance. We need to take the numbers we already have, the means, and make them meaningful.

We need to make Mary's scores in Quiz 1 and Quiz 2 to be comparable to John's. In essence, we need to standardize.

C - The Z-Score[edit]

1 - What is a Z-Score?[edit]

A z-score is the number of standard deviations a particular score deviates from its corresponding mean. In other words, it is the distance, in standard deviations, from the mean.

  • The sign indicates whether the value is above or below the mean.
  • The value indicates the distance, in standard deviation, from the mean.
Computing z-scores, from Dr. Parris' slides in Developing the Z-Score

2 - How is a Z-Score Useful?[edit]

There are several ways that a z-score can be useful. For starters, it is a great way to describe scores in a distribution with a single number, because this number tells you not only how far away is from the mean, but it also tells you how it compares to other scores in that same distribution. A z-score is also great for equating and rescaling entire distributions ,

  • The mean is always zero and the standard deviation is always one.

However, it must be noted that while it transforms the scales of the distribution, it does not change the shape. And perhaps the most useful component to a z-score is that you can compare scores from non-equivalent distributions, much like comparing the scores of two different quizzes.

3 - How Do You Compute a Z-Score?[edit]

Z-Score Objectives[edit]

When computing the z-score, it is important to consider:

  • Center of the distribution for each quiz
  • Meaning of average score for each individual
  • Varability between the two quizzes

Z-Score Computation[edit]

1)Find difference of raw score and quiz mean to measure distance from mean.

2)Measure how spread out each quiz distribution is and how far each score deviates from the mean on average, or the standard deviation.

3)Divide the difference by the standard deviation to standardize the units.

  • This measures how far, on average, each individual is from the center of the distribution.

Therefore, the general formula for computing a z-score is: (raw score - the mean)/ the standard deviation.

Converting Raw and Z-Score[edit]

Z-scores and raw score can be transformed interchangeably when the mean, standard deviation, and either the raw or z-score of the distribution is known.

D - Standardizing Scores in JMP[edit]

1 - Function of Z-Score in JMP[edit]

An example of inputting a fixed formula, slides from Dr. Parris' Using JMP to Standardize Scores.

There are two ways Dr. Julian Parris discussed to calculate the z-score in JMP.

The Standardize Function[edit]

To find the z-scores, follow previous instructions for the mean, except this time click on the standardized column. So,

1) Create and name a new column.

2) Under the new column, right click>formula>statistical>col standardize to input which column to make calculations for.

Save Standardized Function[edit]

1) Under a new column, analyze>distributions>y-columns.

2) Once the display opens, second red triangle>save>standardized

  • This method has no formula and is fixed (Will not change with additional information).

2 - Z-Score Average in JMP[edit]

To find the average z-scores, follow previous instructions for the mean.

Quiz Score Example Revisit[edit]

The average of the z-scores between Mary and John show the significance of utilizing information about the distribution to find a more accurate measure of the raw data. Scaling the two quizzes to be compared relative to the distribution of each quiz gives a better indication of how individuals did in relation to each other.