- We'll decide which write-up you'll hand in for you first "quiz grade"
- #8, 9, 7
- At this point, quizzes become a distinct possibility. Each day
it's possible that you will have a quiz -- at the beginning of class
-- over the material studied one week prior. On Tuesday we finished up
chapter 2: today you should ask questions over chapter 2, and it's all
fair game for a quiz next Tuesday....
Today we'll finish up section 3.1, and perhaps 3.2: therefore 3.1 is a distinct possibility for a quiz next Thursday.
- Here's a histogram plotter for you, useful for experimentation with bin-width.
Measures of central tendency: measures typical (central) values
-
- mean
-
- Also known as the arithmetic average
- center of mass of a histogram
- Notation (see p. 64):
- Suppose that you have n data values.
- population (
-- "mu") versus sample means (
-- "xbar")
- sigma (
) notation
- Sensitive to unusual, extreme values
- Let's compute some means of M&M colors:
orange green red yellow blue brown
-
- median
-
- the middle value: insensitive to unusual, extreme values; but not as good as the mean for purposes of statistical inference.
- In the event that there are an even number of data values, the median is the taken to be mean of the two values "in the middle".
-
- mode
-
- the most "popular" value (or midpoint of the class with largest frequency)
- You may have more than one mode:
- Relationship between measures of central tendency with skewness of a histogram
symmetrically uni-modal (a centered hump): mean and median tend to be the same skewed right (more extremely large values):
as the mean is sensitive to extremes, it chases the tail to the right.skewed left (more extremely small values):
as the mean is sensitive to extremes, it chases the tail to the left. - Example: Computing mean and median for heights of NKU students, using StatCrunch
- Measures of Variability
- A measure of variability is a statistic that describes the spread, or variation, or dispersion in a distribution of data.
-
- range
- is equal to the largest value in the data minus the smallest value
-
- sample variance
-
- the mean of the squared deviations
- Notation - population variance (sigma squared)
versus sample variance (s2)
- Notation, formulas,
examples -- sample versus population
- the mean of the squared deviations
-
- sample standard deviation
-
- is calculated as the square root of the sample variance, and represented by s.
- It is always a non-negative number.
- It has the same units as the data, and as the mean.
- Notation:
- Example for StatCrunch:
shapes of
distributions and variation
- When is
- Variability good?
- Variability bad?
- The empirical rule - the 68/95/99 rule.
This is a rule based on the standard deviation that you'll need to know.
- Based on the standard deviation, which serves as a "typical deviation"
- 68% of the data is within one standard deviation of the mean
- 95% of the data is within two standard deviation of the mean
- 99% of the data is within three standard deviation of the mean
- Works well when the distribution is bell-shaped
- May work okay in any event.
- Example
- Replacing a histogram by a smooth curve....