NAME: ______BLOCK:______DATE:______
3.1 Measures of Central Tendency
Mean, Median and Mode
Exercise 1: Count the letters in each word of this sentence and give the mode. The numbers of letters in the words of the sentence are
5 3 7 2 4 4 2 4 8 3 4 3 4
Mode:
Another average that is useful is the median, or central value, of an ordered distribution. The median is the middle value of a data set.
Exercise 2: Find the median
What do barbecue-flavored potato chips cost? According to Consumer Reports, Vol. 66, No. 5, the prices per ounce in cents of the rated chips are
19 19 27 28 18 35
Median:
Case: n is even.
Exercise 3: To graduate, Linda needs at least a B in biology. She did not do very well on her first three tests; however, she did well on the last four. Here are her scores:
58 67 60 84 93 98 100
Compute the mean and determine if Linda’s grade will be a B (80 to 89 average) or a C (70 to 79 average)
Solution:
Exercise 4: The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?
872 432 397 427 388 782 397
Solution:
Exercise 5: The unit load of 40 randomly selected students from a college shown below. Find the mean, median and mode. Optional: Use TI-84 graphing calculator.
17 / 12 / 14 / 17 / 13 / 16 / 18 / 20 / 13 / 1212 / 17 / 16 / 15 / 14 / 12 / 12 / 13 / 17 / 14
15 / 12 / 15 / 16 / 12 / 18 / 20 / 19 / 12 / 15
18 / 14 / 16 / 17 / 15 / 19 / 12 / 13 / 13 / 15
Mean
Median
Mode
You can find the mean, median, sum, and other important numbers associated
with a list by pressing 2nd [LIST] and arrowing over to MATH. Always specify the
list name and close the parentheses.
1:min( the minimum value in a list
2:max( the maximum value in a list
3:mean( the mean of the list entries
4:median( the median of the list entries
5:sum( the sum of the list entries
Exercise 6: Barron’s Profiles of American Colleges, 19th Edition, lists average class size for introductory lecture courses at each of the profiled institutions. A sample of 20 colleges and universities in California showed class sizes for introductory lecture courses to be
14 / 20 / 20 / 20 / 20 / 23 / 25 / 30 / 30 / 30 / 35 / 35 / 35 / 40 / 40 / 42 / 50 / 50 / 80 / 80a) Compute the mean b) compute a 5% trimmed mean
Exercise 7: You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A? Repeat the calculation using the 1-Variable Stats function in your graphing calculator. Solution:
Category / Score x / Weight w / xwTEST / 86 / 50%
Midterm / 96 / 15%
Computer / 98 / 10%
Final / 82 / 20%
HW / 100 / 5%
TOTAL / 100%
Exercise 8: Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session.
Lower Limit / Upper Limit / Midpoint x / Frequency f / x * f7 / 18 / 12.5 / 6
19 / 30 / 24.5 / 10
31 / 42 / 36.5 / 13
43 / 54 / 48.5 / 8
55 / 66 / 60.5 / 5
67 / 78 / 72.5 / 6
79 / 90 / 84.5 / 2
TOTAL / 50
Sort and 1-Variable Statistics
i. Enter the data into a list, say : STAT / 1: Edit
ii. Compute 1-variable statistics on the data list
STAT / CALC / 1: 1-Var Stats
iii. Find the mode by sorting the list in ascending order
STAT / 2: SortA
Weighted Average on the TI-83/84
Enter the data and the corresponding weights into two lists and run 1-Variable Statistics:
STAT / CALC / 1: 1-Var Stats Ldata, Lfrequency or
STAT / CALC / 1: 1-Var Stats Ldata, Lweight
1
NAME: ______BLOCK:______DATE:______
3.2 Measures of Variance (or Dispersion)
Range
Range is the difference between the maximum and minimum data entries in the set. The data must be quantitative. Range = (Max. data entry) – (Min. data entry)
Deviation
Deviation is the difference between the data entry, x, and the mean of the data set.
Exercise 1: A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries.
x / x - µ41
38
39
45
47
41
44
41
37
42
µ = / 41.5
Population Notation, Measures and Formulas
a. Population size is denoted N
b. Population mean, (read “mew”):
c. Population standard deviation =
d. Population variance =
Sample Measures of Variance (Dispersion)
1. Range = high value - low value
2. Sample Standard Deviation =
where n = sample size, x = sample mean
3. Sample Variance =
Exercise 2: A corporation hired 10 graduates. The starting salaries for each graduate were stated above. Find the population standard deviation of the starting salaries.
x / x - µ / (x - µ) 241
38
39
45
47
41
44
41
37
42
µ =
σ2
σ
Exercise 3: The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries.
Sample Variance:
Sample Standard Deviation:
Technology: Use the graphing calculator to find the sample mean and standard deviation.
Fact: Standard deviations can be compared only when the units are the same and/or the populations are similar.
Coefficient of Variation
The Coefficient of Variation is a unit-less measure of variance and expresses the standard deviation as a percent of the mean.
Sample Coefficient of Variation:
Population Coefficient of Variation:
Exercise 4: The Trading Post on Grand Mesa is a small, family-run store in a remote part of Colorado. It has just eight different types of spinners for sale. The prices (in dollars) are:
2.10 1.95 2.60 2.00 1.85 2.25 2.15 2.25
Since the Trading Post has only eight different kinds of spinners for sale, we consider the eight data values to be the population.
a) Use a calculator with appropriate statistics keys to verify
that for the Trading Post data, and m » $2.14 and s » $0.22.
b) Compute the coefficient of variation
Chebyshev’s Theorem
For any set of data and for any constant k > 1, the percent of the data values that must lie within k standard deviations on either side of the mean is at least
Chebyshev’s Theorem gives the Minimum Percentage of Data that lie within k Standard Deviations of the Mean
k / 2 / 3 / 4 / 5 / 10/ 75% / 88.9% / 93.8% / 96% / 99%
Exercise 5:
A newspaper periodically runs an ad in its own advertising section offering a free month’s subscription. In this way, management can get an idea of how many people read the classifieds. Over a period of two years the mean number of responses was with a sample standard deviation of s = 30.
a. What is the smallest percentage of data we expect to fall within 2 standard deviations of the mean (i.e. between 465 and 585
b. Determine the interval from A to B about the mean in which 88.9% of the data fall.
c. What is the smallest percent of respondents to the ad that falls within 2.5 standard deviation of the mean?
d. What is the interval from A to B from part c. Explain its meaning in this application.
Mean & Standard Deviation of Grouped Data
1. Make a frequency distribution table [from the histogram if necessary].
a. Compute the class midpoint for each class; this is the “best guess” of each data value in the class. Place the class midpoints in list L1.
b. Place the corresponding frequency of each class in list L2.
2. Compute 1-variable statistics on L1 and L2
STAT / CALC / 1: 1-Var Stats Ldata, Lfrequency or
STAT / CALC / 1: 1-Var Stats Ldata, Lweight
Exercise 6:
You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set.
1 / 3 / 1 / 1 / 1 / 1 / 1 / 6 / 0 / 11 / 2 / 2 / 1 / 0 / 3 / 6 / 6 / 1 / 2
1 / 1 / 0 / 0 / 0 / 2 / 3 / 0 / 1 / 1
1 / 5 / 0 / 3 / 6 / 4 / 1 / 1 / 2 / 2
3 / 0 / 3 / 1 / 1 / 0 / 3 / 0 / 2 / 4
Sample Mean: ______
x / f / (x - x-bar) / (x – x-bar)^2 / (x – x-bar)^2 * f0 / 10
1 / 19
2 / 7
3 / 7
4 / 2
5 / 1
6 / 4
TOTAL
Sample Standard Deviation: ______
3.3 Percentiles and Box-and-Whisker Plots
Percentiles
A percentile ranking gives a rank relative to all other data values. For whole numbers P, the Pth percentile of a distribution is a value such that P% of the data fall at or below that value. Thus, the median value is the same as the 50th percentile value.
Exercise 1
Suppose you challenge freshman composition by taking an exam.
a. If your score was in the 89th percentile, what percentage of scores was at or below your score?
b. If the scores ranged from 0 to 100 and your raw score was 95, does that mean that your score is at the 95th percentile?
Quartiles, Interquartile Range, 5-Number Summary
Quartiles are percentiles that divide the data into fourths. The first quartile Q1 is the 25th percentile, the second quartile Q2 is the median, and the third quartile Q3 is the 75th percentile. The interquartile range is Q3 – Q1; it is a measure of how spread-out the middle 50% of the data is. The 5-number summary is the lowest value, Q1, median, Q3, and the highest value.
Exercise 2
The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores:
13 9 18 15 14 21 7 10 11 20 5 18 37 16 17
Solution:
Exercise 3 Find the interquartile range of the test scores.
Solution:
Box-and-Whisker Plots
1. Enter the data into a list and run 1-variable statistics to find the 5-number summary: lowest value, Q1, median, Q3, highest value
2. Draw an axis (horizontal or vertical) and scale it to include the lowest and highest values.
3. To the right of (or above) the axis draw a box around the interquartile range (from Q1 to Q3) and a line inside the box at the median.
4. Draw whiskers from Q1 to the lowest value, and from Q3 to the highest value.
Exercise 4
Draw a box-and-whisker plot that represents the 15 test scores.
Exercise 5
Use your box-and-whisker plot to write an interpretive statement:
1