Ways to look at the data Number of hurricanes that occurred each year from 1944 through 2000 as reported by Science magazine Histogram Dot plot Box plot

3 Characteristics of data Shape Center Spread

Shape of the data – Symmetric The age of all US Presidents at the time they took office Notice that this distribution has only one mode

Shape of the data – Bimodal The winning times in the Kentucky Derby from 1875 to the present. Why two modes?

Shape of the data – Bimodal The winning times in the Kentucky Derby from 1875 to the present. Why two modes? The length of the track was reduced from 1.5 miles to 1.25 miles in The race officials thought that 1.5 miles was too far.

Shape of the data – skewed Data for two different variables for all female heart attack patients in New York state in one year. One is skewed left; the other is skewed right. Which is which? LEFTRIGHT

Center and Spread of Data Maximum Q3 Median Q1 Minimum 100 th percentile 75 th percentile 50 th percentile 25 th percentile 0 th percentile These numbers are called the 5 number summary. The median measures the center of the data. Q3 – Q1 = Interquartile range (IQR) measures the spread.

Symbols: s 2 = Sample Variance s = Sample Standard Deviation  2 = Population Variance (Pop. St. Dev. Squared)  = Population Standard Deviation (Sq. Root of Variance) REMEMBER-The Variance is the SD squared! And the SD is the Sq. root of the Variance! x = Mean Symbols --

The normal distribution and standard deviations The total area under the curve is 1. In a normal distribution: 34% 13.5% 2.35%

The normal distribution and standard deviations Approximately 68% of scores will fall within one standard deviation of the mean In a normal distribution:

The normal distribution and standard deviations Approximately 95% of scores will fall within two standard deviations of the mean In a normal distribution:

The number of points that one standard deviations equals varies from distribution to distribution. On one math test, a standard deviation may be 7 points. If the mean were 45, then we would know that 68% of the students scored from 38 to Points on Math Test Points on a Different Test On another test, a standard deviation may equal 5 points. If the mean were 45, then 68% of the students would score from 40 to 50 points. 2.35% 13.5% 34% 34% 13.5% 2.35%

Using standard deviation units to describe individual scores sd1 sd2 sd-2 sd What score is one sd below the mean?90 What score is two sd above the mean? 120 Here is a distribution with a mean of 100 and standard deviation of 10:

Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and standard deviation of 10: sd1 sd2 sd-2 sd How many standard deviations below the mean is a score of 90? 1 2 How many standard deviations above the mean is a score of 120?

Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and standard deviation of 10: sd1 sd2 sd-2 sd What percent of your data points are < 80? 2.50% 84% What percent of your data points are > 90?

Types of Sampling: Self-selected Sample This methods allows the sample to choose themselves by responding to a general appeal (volunteering to be surveyed). Examples of Self-selected Sample: a call- in radio poll, an internet poll on a website Problems with Self-selected samples: bias – because people with strong opinions on the topic (especially negative opinions) are most likely to respond.

Convenience Sampling In a convenience sample individuals are chosen because they are easy to reach. Example: People conducting a survey go to the mall and stop people who are shopping. This is convenient for the person doing the survey but does not guarantee that the sample is representative of the population of the study. Convenience sampling also involves bias on the part of the interviewer.

Random Samples A random sample of size “n” individuals from the population chosen in such a way that every set of “n” individuals has an equal chance to be the sample selected. Example: Putting everyone’s name in a hat and drawing 3 names to participate in the study.

Systematic Sample When a rule is used to select members of the population. Ex. Every third person on an alphabetized list

Stratified Random Sample To select a stratified random sample, first divide the population into groups of similar individuals, called STRATA. Then choose a separate sample in each strata and combine these to form the full sample. Common example would be separating by gender or race first, then selecting samples from each group.