Z-Scores and Percentiles: Crash Course Statistics #18

CrashCourse
30 May 201810:54
EducationalLearning
32 Likes 10 Comments

TLDRThis video explains how to compare things measured on different scales using z-scores. It first shows how to adjust and standardize SAT and ACT scores so they can be directly compared. It then explains how z-scores allow you to compare any normal distribution, like athlete performance or apple weights, in terms of standard deviations from the mean. These normalized scores make it easy to calculate percentiles and determine if a score is extremely high or low compared to a population.

Takeaways
  • 😀 Z-scores allow you to compare things measured on different scales by standardizing the distributions.
  • 📊 To calculate a z-score, subtract the mean from the raw score, then divide by the standard deviation.
  • 🔢 Higher z-scores indicate values further from the mean; lower z-scores are closer to the mean.
  • 🍎 Z-scores can tell you percentiles - the percentage of values below a certain score.
  • 📈 Percentiles are useful for comparing yourself to a population distribution.
  • 🏆 Z-scores help compare athletes from different sports by seeing how far above average their stats are.
  • 🤔 There's no definitive threshold for determining if a value is extremely high/low based on z-scores.
  • 🤯 You can use z-scores to determine if a value is likely from a certain distribution.
  • 📉 Standardizing distributions centers them around a mean of 0.
  • 😎 Z-scores allow comparisons of 'apples to oranges'.
Q & A

  • What are two examples given of how statistics can be used to compare things?

    -The video gives two examples of using statistics for comparison: 1) Comparing your GPA to the mean or median GPA. 2) Using standard deviation to see if the amount of time people spend on social media is similar or varies a lot.

  • Why can standardization help when comparing things measured on different scales?

    -Standardization converts different scales into a common scale (z-scores) with a mean of 0 and standard deviation of 1. This allows for direct comparison of things like test scores that are originally on different scales.

  • How can you compare an SAT score to an ACT score?

    -To compare SAT and ACT scores: 1) Subtract the mean score of each test to center them around 0. 2) Divide the centered scores by the standard deviation of each test to convert to z-scores. The z-scores can then be directly compared.

  • What do percentiles tell you?

    -Percentiles tell you the percentage of the population that has a score lower than yours. For example, being in the 90th percentile means 90% of scores are below yours.

  • How can you find the score needed to be in a certain percentile?

    -To find the score for a percentile: 1) Find the z-score for that percentile. 2) Multiply the z-score by the standard deviation and add the mean to convert back to the original scale.

  • How can z-scores help determine if a value is extremely high or low?

    -A z-score shows how many standard deviations a value is from the mean. A very high positive or negative z-score indicates an extreme value - though how extreme is somewhat arbitrary.

  • What factors determine whether a high z-score means a value is not from the distribution?

    -To decide if a high z-score means a value is not from the distribution, consider: 1) How extreme the z-score is (e.g. 3.5 vs. 1.65) 2) What the other possible distributions are.

  • How could z-scores be used to compare athletes to determine the Greatest of All Time?

    -To compare athletes across sports, their key stats could be converted to z-scores based on averages in their sport. The athlete with the highest z-score might be considered the Greatest of All Time.

  • What is the purpose of converting scores to z-scores?

    -The main purpose of z-scores is to standardize different scales to a common one with mean 0 and standard deviation 1. This allows for direct numerical comparison of scores from different distributions.

  • What do higher positive or negative z-scores indicate about a value?

    -Higher positive z-scores indicate a value far above the mean, while highly negative z-scores indicate a value far below the mean. Extreme z-scores in either direction indicate rare or unusual values.

Outlines
00:00
📺 Introduction to comparing things with statistics and standardizing scores

This first paragraph introduces the video and the concept of using statistics to compare things. It gives examples like GPA and time spent on social media. It then explains that sometimes you want to compare things measured on different scales, which is where standardizing scores by converting to z-scores comes in. Examples are given of SAT and ACT scores, which are standardized so they can be directly compared.

05:03
🚀 Converting test scores to z-scores to enable comparison

This paragraph provides a specific example of two students - Tony and Maia - who took different college admission tests (SAT and ACT). It explains the process of standardizing their scores into z-scores by first centering them based on the mean, and then dividing by the standard deviation. This puts the scores on the same scale so they can be easily compared to evaluate who performed better relative to their test.

10:05
🥇 Using percentiles and z-scores to determine extreme scores

The last paragraph explains how percentiles can be calculated from z-scores to determine what percentage of people score below a certain value. An example is provided related to qualifying for a video game competition. The concept of what constitutes an extreme z-score is also discussed, along with how to interpret whether something with a very high or low z-score is likely to be from a particular distribution.

Mindmap
Keywords
💡Standardization
Standardization refers to the process of adjusting different sets of data to a common scale, allowing for meaningful comparisons between them. In the video, this concept is crucial for comparing scores from different tests, such as the SAT and ACT, which are on different scales. By standardizing scores, we can directly compare the performance of individuals on these different tests, making it easier to understand relative performance levels.
💡Z-score
A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. The video uses z-scores to compare SAT and ACT scores by converting them into a standard scale where 0 represents the mean score and 1 indicates one standard deviation from the mean. This makes it possible to directly compare scores from different tests by showing how far each score is from the average in terms of standard deviations.
💡Percentiles
Percentiles are a statistical measure indicating the value below which a given percentage of observations in a group of observations fall. The video explains how percentiles can be used to understand a student's performance relative to peers. For example, being in the 80th percentile means a score is higher than 80% of the scores. This concept is used alongside z-scores to determine how an individual's standardized test score compares to the wider population.
💡Standard deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that most of the numbers are close to the mean (average) value, while a high standard deviation means that the values are spread out over a wider range. The video discusses using standard deviation in the process of standardizing test scores, which helps in comparing scores from tests like the SAT and ACT by scaling them relative to how spread out each test's scores are.
💡Normal distribution
Normal distribution, or the bell curve, is a type of distribution where most observations cluster around the central peak and the probabilities of values further away from the mean taper off equally in both directions. The video references the normal distribution when discussing the calculation of z-scores and percentiles, as these concepts assume the data follows a normal distribution for their calculations to be meaningful.
💡Mean
The mean, or average, is the sum of all values in a set divided by the number of values. The video highlights the importance of the mean in the standardization process, where scores from tests like the SAT and ACT are adjusted by subtracting the mean of each test to center the distributions around zero. This step is essential for comparing different sets of data on a common scale.
💡SAT and ACT
The SAT and ACT are standardized tests used for college admissions in the United States, each measured on different scales. The video uses these tests as an example to illustrate the challenge of comparing scores across different scales and the application of standardization and z-scores to make these scores comparable, emphasizing the importance of these statistical concepts in educational contexts.
💡Comparing distributions
This concept involves analyzing two or more sets of data to understand their differences and similarities. The video applies this idea in the context of standardizing test scores, showing how to adjust and compare distributions of SAT and ACT scores. By centering scores around zero and scaling by standard deviation, it becomes possible to compare performances on these different tests directly.
💡Adjusted scores
Adjusted scores are the result of modifying original scores to make them comparable across different scales. In the video, adjusted scores are created by first centering SAT and ACT scores around zero (by subtracting the mean) and then scaling them by standard deviation. This process yields scores that can be directly compared, regardless of the original scale of measurement.
💡Extreme scores
Extreme scores are significantly higher or lower than the average scores in a distribution. The video discusses how identifying extreme scores, using z-scores, can indicate performances that are significantly above or below the norm. This concept is illustrated with examples, including the humorous scenario of guessing whether an unusually heavy object at a fair is an apple, highlighting how statistical tools help make informed judgments about outliers.
Highlights

First significant research finding

Introduction of new theoretical model

Description of innovative experimental method

Transcripts

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