1 / 25

Chapter 6

Chapter 6. The Standard Deviation as a Ruler and the Normal Model . Problem. The result of a 50-point test looks bad, and I Knew that I made the test too hard compared to the test from last semester. So I decide to add 5 points to each student’s score. Ann: 40 pts Bob: 25 pts

fern
Download Presentation

Chapter 6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6 The Standard Deviation as a Ruler and the Normal Model

  2. Problem • The result of a 50-point test looks bad, and I Knew that I made the test too hard compared to the test from last semester. So I decide to add 5 points to each student’s score. • Ann: 40 pts • Bob: 25 pts • Carol: 37 pts • Don: 14 pts • Eva: 14 pts

  3. Shifting Data • Adding (or subtracting) a constant to every data value adds (or subtracts) the same constant to measures of position. • Adding (or subtracting) a constant to each value will increase (or decrease) measures of position: center, percentiles, max or min by the same constant. • Its shape and spread - range, IQR, standard deviation - remain unchanged.

  4. Shifting Data (cont.) • The following histograms show a shift from men’s actual weights to kilograms above recommended weight:

  5. Problem(cont.) • After I “curve”, I realize that my syllabus indicate that each test is worth 150 points. How can I fix it to align with the policy listed on the syllabus? • Student Ann: 45 pts  50 pts • Student Bob: 30 pts  35 pts • Student Carol: 42 pts  47 pts • Student Don: 19 pts  24 pts • Student Eva: 19 pts  24 pts

  6. Rescaling Data • When we multiply (or divide) all the data values by any constant, all measures of position (such as the mean, median, and percentiles) and measures of spread (such as the range, the IQR, and the standard deviation) are multiplied (or divided) by that same constant.

  7. Rescaling Data (cont.) • The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data:

  8. Example • Scores on the ACT college entrance exam in a recent year were roughly normal, with mean 21.2 and standard deviation 4.8. John scores 27 on the ACT. • Scores on the SAT Reasoning college entrance exam in the same year were roughly normal, with mean 1511 and standard deviation 194. Susan scores 1718 on the SAT. • Who has a higher score? Which student is considered a better student?

  9. The Standard Deviation as a Ruler • The trick in comparing very different-looking values is to use standard deviations as our rulers. • The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group. • As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

  10. We compare individual data values to their mean, relative to their standard deviation using the following formula: We call the resulting values standardized values, or z-scores. Standardizing with z-scores

  11. Standardizing with z-scores (cont.) • Standardized values have no units. • z-scores measure the distance of each data value from the mean in standard deviations. • A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

  12. Benefits of Standardizing • Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean. • Thus, we can compare values that are measured on different scales, with different units, or from different populations.

  13. Back to z-scores • Standardizing data into z-scores shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation (SD). • Standardizing into z-scoresdoes not change the shape of the distribution. • Standardizing into z-scoreschanges the center by making the mean 0. • Standardizing into z-scoreschanges the spread by making the standard deviation 1.

  14. Normal Model • A useful family of models for unimodal, symmetric distributions, usually represented by a bell-shaped curves, is called the Normal model. • Biological measures (like height, weight, heart rate, head circumference, etc.) are often normally distributed. • We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ.

  15. Normal Model • Summaries of data, like the sample mean and standard deviation, are written with Latin letters. Such summaries of data are called statistics. • When we standardize Normal data, we still call the standardized value a z-score, and we write • Once we have standardized, we need only one model, the N(0,1) model is called the standard Normal model(or the standard Normal distribution).

  16. The 68-95-99.7 Rule • Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean. • It turns out that in a Normal model: • about 68% of the values fall within 1 SD of the mean • about 95% of the values fall within 2 SDof the mean • about 99.7% of the values fall within 3 SD of the mean.

  17. Working with Normal Models • When we use the Normal model, we are assuming the distribution is Normal. • A newborn baby weighs 8.5 lbs, is it normal? • We cannot check this assumption in practice, so when we have the actual data, we make a histogram of the distribution and check the Nearly Normal Condition: • The shape of the data’s distribution is unimodal? • Bell-shaped curve? • Roughly symmetric?

  18. Finding Normal Percentiles by Hand • When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles. • Table Z in Appendix D provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.

  19. Finding Normal Percentiles by Hand (cont.) • Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. • The figure shows us how to find the area to the left when we have a z-score of 1.80:

  20. From Percentiles to Scores: z in Reverse • Sometimes we start with areas and need to find the corresponding z-score or even the original data value. • Example: What z-score represents the first quartile in a Normal model?

  21. From Percentiles to Scores: z in Reverse (cont.) • Look in Table Z for an area of 0.2500. • The exact area is not there, but 0.2514 is pretty close. • This figure is associated with z = -0.67, so the first quartile is 0.67 standard deviations below the mean.

  22. Example • The distribution of scores on tests such as the SAT college entrance examination is close to normal. Scores on each of the three sections (math, critical reading, writing) of the SAT are adjusted so that the mean score is about 500 and the standard deviation is about 100. • What percent of scores fall between 200 and 800? • What percent of scores are above 700? • How high must a student score to fall in the top 25%? • What proportion of SAT scores are above 640?

  23. Example • The scale of scores on an IQ test is approximately normal with mean 100 and standard deviation 15. The organization MENSA, which calls itself “the high IQ society,” requires an IQ score of 130 or higher for membership. What percent of adults would qualify for membership? • a) 95% b) 5% c) 2.5% d) 17%

  24. Example • Scores on the ACT college entrance exam in a recent year were roughly normal, with mean 21.2 and standard deviation 4.8. John scores 27 on the ACT. • Scores on the SAT Reasoning college entrance exam in the same year were roughly normal, with mean 1511 and standard deviation 194. Susan scores 1718 on the SAT. • Who has a higher score?

  25. Homework Assignment Page 147 – 152 Problem # 3, 5, 9, 11, 15, 19, 27, 29, 33, 35, 37, 39, 41, 43, 49, 53, 57.

More Related