Measures of Relative Standing: z-Scores

Summary

The script delves into z-scores, a measure of relative standing that ranks data points by their distance from the mean in standard deviation units. It explains how a negative z-score indicates a value below the mean, while a positive suggests above. Z-scores are crucial for comparing observations across different datasets. The script also touches on standardized datasets, where all values are converted to z-scores, resulting in a mean of zero and a standard deviation of one. It concludes with the application of z-scores in identifying outliers and significant observations, referencing Chebyshev's and the Empirical Rule for further interpretation.

Takeaways

• 📊 The z-score (or standard score) measures an observation's relative standing in a dataset by calculating z = (x_i - μ) / σ, where x_i is the observation, μ is the mean, and σ is the standard deviation.
• 🔢 A negative z-score indicates an observation below the mean, while a positive z-score indicates an observation above the mean.
• 📉 The z-score represents the number of standard deviations an observation is from the mean, providing insight into its relative position within the dataset.
• 🎓 Historical example: Lincoln's age at inauguration (52) has a z-score of -0.46, indicating it's 0.46 standard deviations below the mean, while Eisenhower's age (62) has a z-score of 1.08, showing it's 1.08 above the mean.
• 🆚 Z-scores allow for the comparison of relative standings between observations from different distributions, as demonstrated by comparing Arthur's and Bethany's exam scores.
• 📚 A standardized dataset is created by converting all observations to their z-scores, resulting in a mean of zero and a standard deviation of one.
• 📉 Outliers can be identified with z-scores: observations with z-scores less than -3 or greater than 3 are often considered outliers.
• 📈 Significant observations are defined by z-scores: those less than -2 are significantly low, and those greater than 2 are significantly high.
• 📊 Chebyshev's Rule states that in any dataset, at least 75% of observations fall within a z-score range of -2 to 2, and at least 89% fall within -3 to 3.
• 📈 The Empirical Rule applies to bell-shaped datasets, stating that approximately 68% of observations have z-scores between -1 and 1, 95% between -2 and 2, and 99.7% between -3 and 3.

Q & A

What is a z-score and what is another term used for it?

A z-score is a measure of relative standing that indicates how many standard deviations an observation is from the mean. It is calculated using the formula z = (x_i - mu) / sigma, where x_i is the observation, mu is the mean, and sigma is the standard deviation. Another term used for z-score is 'standard score'.

What is the average age of all presidents at inauguration and the standard deviation?

The average age of all presidents at inauguration is fifty-five, and the standard deviation is six and a half.

What are the z-scores for Lincoln and Eisenhower based on their ages at inauguration?

Lincoln's z-score is -0.46, indicating he was 0.46 standard deviations below the mean. Eisenhower's z-score is 1.08, indicating he was 1.08 standard deviations above the mean.

What does a negative z-score signify in terms of an observation's position relative to the mean?

A negative z-score signifies that the observation is below the mean of the dataset.

What does a positive z-score signify in terms of an observation's position relative to the mean?

A positive z-score signifies that the observation is above the mean of the dataset.

What does the z-score of an observation represent in terms of its distance from the mean?

The z-score of an observation represents the number of standard deviations that the observation is away from the mean.

What does a z-score of three or more indicate about an observation's relative standing?

A z-score of three or more indicates that the observation is larger than most of the other observations in the dataset.

What does a z-score of negative three or less indicate about an observation's relative standing?

A z-score of negative three or less indicates that the observation is smaller than most of the other observations in the dataset.

What does a z-score near zero suggest about an observation's position in the dataset?

A z-score near zero suggests that the observation is located near the mean of the dataset.

In the example given, who scored relatively better on their exams, Arthur or Bethany, and why?

Bethany scored relatively better than Arthur despite having a lower exam score because her z-score was higher (3 compared to Arthur's 2), indicating she performed better relative to her class's mean and standard deviation.

What is a standardized dataset and how is it created?

A standardized dataset is a set consisting of the z-scores of all observations. It is created by replacing each value in the original dataset with its corresponding z-score, resulting in a set where the mean is always zero and the standard deviation is always one.

How can the rules from the previous section be rephrased using z-score language to identify outliers?

Using z-score language, any observation with a z-score less than negative three or greater than three is considered an outlier, which is a rephrasing of the three standard deviation rule.

According to Chebyshev's Rule, what percentage of observations in any dataset will have z-scores between negative two and two?

According to Chebyshev's Rule, at least 75% of observations in any dataset will have z-scores between negative two and two.

By the Empirical Rule, what percentage of observations in a bell-shaped dataset will have z-scores between negative one and one?

By the Empirical Rule, approximately 68% of observations in a bell-shaped dataset will have z-scores between negative one and one.