D E C E M B E R 2 , 2 0 1 3

MATH 010: 10.1 & 10.2STATISTICS

10.1 ORGANIZING DATA

• Know how to make a frequency distribution using a set of data.

• Set of collected scores on Exam #3 (this is our population):

• The range is the difference between the least and greatest values in the set• 94 - 14 = 80

14 62 74 77 86 89

47 68 74 79 87 94

52 70 76 80 87 94

61 70 77 83 89 94

10.1 FREQUENCY DISTRIBUTION

• Create a frequency distribution with a class width of 8.

• To create intervals, add the class width to the lowest value.

• 14 + 8 = 22 so first interval is “14-22”• The next intervals are “23-31”, “32-40”, “41-49”, “50-

58”, “59-67”, “68-76”, “77-85”, “86-94”• 9 intervals total, so there are 9 classes.

14 62 74 77 86 8947 68 74 79 87 9452 70 76 80 87 9461 70 77 83 89 94

FREQUENCY DISTRIBUTION

Classes Frequency

14-22 1

23-31 0

32-40 0

41-49 1

50-58 1

59-67 2

68-76 6

77-85 5

86-94 8

14 62 74 77 86 8947 68 74 79 87 9452 70 76 80 87 9461 70 77 83 89 94

HISTOGRAM

14-22 23-31 32-40 41-49 50-58 59-67 68-76 77-85 86-940123456789

Exam #3 Collected Score Distribution

• Question: What percentage of students scored between 86 and 94 on the exam?

• 24 is the number of total scores (base). 8/24 = 0.333 = 33.3%

• Know how to calculate percentages based on a histogram.

WHAT HISTOGRAMS TELL YOU

• Histogram A is skewed to the left, so there are more values on the low end of the range.• Histogram B is skewed to the right, so there

are more values on the high end of the range.• Histogram C is a normal distribution, which

means the values are mostly near the median.

10.2 TYPES OF AVERAGES

• Mean: Add all values, then divide the sum by the number of values. Sum of values = 1784

• Mean score = 1784 / 24 = 74.333• Median: Put values in numerical order, then find the value

in the middle of the list.• If there is an even number of values, the median is halfway

between the two middle values.• Median score = 77• Mode: The most commonly occurring value.• Mode score = 94 (occurs three times)

14 62 74 77 86 8947 68 74 79 87 9452 70 76 80 87 9461 70 77 83 89 94

WHICH TYPE OF AVERAGE TO USE?

• What does each type of average tell us?

• Which average best represents the set of scores?

Mean: 74.3

Median: 77

Mode: 94

BOX-AND-WHISKERS PLOT

• Know how to make a box-and-whiskers plot.• First find the median (77)• Now, split the list of values in half.• Q1 is the median of the first half.• Even number of values, so Q1 is halfway between 68 and 70.• Q1 = 69• Q3 is the median of the second half.• Q3 is halfway between 87 and 87• Q3 = 87

14 62 74 77 86 8947 68 74 79 87 9452 70 76 80 87 9461 70 77 83 89 94

BOX-AND-WHISKERS PLOT

Smallest

value

Largest value

Q1 Q3Median

http://math.andyou.com/tools/boxandwhisker.html

Need to know how to create box-and-whiskers plots with any given set of data

WHAT BOX PLOTS TELL YOU

CORRELATION (NOT IN BOOK)

• A correlation is a relationship between two variables. (poll)• A positive correlation means as one variable increases, the

other variable increases• Example: There is a positive correlation between gun

ownership and homicide rate in a given state. (link)• This means the higher the rate of gun ownership, the higher the

homicide rate.

• A negative correlation means as one variable increases, the other variable decreases.

• Example: There is a negative correlation between couples’ relationship satisfaction and the frequency of texting about relationship problems. (link)• This means if the frequency of texting about relationship problems is

higher, relationship satisfaction is lower, and vice versa.

• Correlation does NOT imply causation!• Just because there is a relationship between two variables, it does not

mean that one caused the other.

HOW STRONG IS THE CORRELATION?

• is called the coefficient of determination.

• It shows the strength of correlation (how well it can fit into a linear model)

• Between 0 (no correlation) and 1 (total correlation)

• Relationships are only considered significant when is greater than 0.9

CORRELATION DOES NOT IMPLY CAUSATION!

• In this example, there is a negative correlation between highway fatality and lemon imports.

• is high, so there is a very strong correlation.

• Does this mean that if we import more lemons, there will be fewer highway fatalities?

QUIZ

1. Interpret the box plot below. What does it say about the distribution of test scores?

2. Find the mean, median, and mode of this set: {0, 1, 2, 2, 3, 3, 3, 4}

3. If there is a negative correlation between hours spent in class and level of satisfaction with the course, what does that mean? What is NOT implied?