Thinking Mathematically

1
Thinking Mathematically

• Chapter 12
• Statistics

2
Thinking Mathematically

• Section 1
• Sampling, Frequency Distributions, and Graphs

3
Statistics

• Statistics is the science of data. This
  involves collecting, classifying, summarizing,
  organizing, analyzing, and interpreting numerical
  information.

4
Types of Statistics

• Descriptive Statistics utilizes numerical and
  graphical methods to look for patterns in a data
  set, to summarize the information revealed in a
  data set, and to present that information in a
  convenient form.
• Example analysis of scores on an exam to see how
  hard it was and how you might "curve" the grades

• Inferential Statistics utilizes sample data to
  make estimates, decisions, predictions, or other
  generalizations about a larger set of data.
• Example looking at census data to determine what
  the make-up of the population will be in 2030.

5
Random Samples

• A random sample is a sample obtained in such a
  way that every element in the population has an
  equal chance of being selected for the sample.
• If we want to select a random sample from a large
  city to determine how the citys citizens feel
  about casino gambling we might
• randomly select neighborhoods of the city and
  then
• randomly survey people within the selected
  neighborhoods.
• If we only select specific neighborhoods or the
  first 200 people we find in the telephone
  directory, then not everyone has an equal chance
  of being selected.

6
Describing Qualitative Data
class frequency
class

• A class is one of the categories into which
  qualitative data can be classified.
• The class frequency is the number of observations
  in the data set falling in a particular class.

7
Histogram

• A histogram is like a bar graph in that the
  vertical axis gives the proportion (or relative
  frequency) for each interval of data while the
  horizontal axis is divided into specified
  intervals of equal width known as measurement
  classes.
• However, in a histogram, each column shares a
  side or touches while in a bar graph each column
  is separated.

8
Histogram
9
Frequency Polygon

• A line graph called a frequency polygon can also
  be used to visually convey information.
• The axes are labeled just like those in a
  histogram. Once a histogram has been
  constructed, put a dot at the top of each
  rectangle at its midpoint.
• Connect each of these midpoints with a straight
  line.
• Finally, draw each endpoint down to touch the
  horizontal axis.

10
Frequency Polygon
11
Stem-and-Leaf Display

• Two columns are created, one for the stem and one
  for the leaf. The place value for the stem is
  determined for the left column and the following
  place value in each piece of data will be written
  next to the appropriate stem in the leaf column.
  As an example, if the data value is 32, the 3 may
  be designated as the stem in which case the 2
  would be the leaf. If the data point is 5.8, the
  ones place may be the stem and the tenths place
  the leaf so that the 5 would be in the stem
  column with the 8 next to it in the leaf column.

12
Stem-and-Leaf Display
7
7
4
3
5
2
2
3
13
Thinking Mathematically

• Section 2
• Measures of Central Tendency

14
The Mean

• The mean of a set of quantitative data is the sum
  of the measurements divided by the number of
  measurements contained in the data set. (the
  average)

15
The Mean

• The mean is the sum of the data items divided by
  the number of items.

16
Computing the Mean
The sum of these numbers is 2907
There are 40 numbers
The mean is 2907/40 72.675
17
Calculating the Mean from a Frequency Distribution
18
Calculating the Mean from a Frequency Distribution
(85)(3) 255 (75)(5) 375 (70)(6)
420 (55)(3) 165 (25)(1) 25
1240
18
Mean 1240/18 68.9
19
Computing a GPA (weighted average)
9.99 5.33 16.00 9.00 9.32
49.64
16
GPA 49.64/16 3.10
20
The Median

• To find the median of a group of data items,
• 1. Arrange the data items in order, from
  smallest to largest.
• 2. If the number of data items is odd, the
  median is the item in the middle of the list.
• 3. If the number of data items is even, the
  median is the mean of the two middle data items.

21
Position of the Median

• If n data items are arranged in order, from
  smallest to largest, the median is the value in
  the

13/2 6.5, so median falls between 6th and 7th
position.
median (7060)/2 65
22
Position of the Median

• If n data items are arranged in order, from
  smallest to largest, the median is the value in
  the
• position.

14/2 7, so median is the 7th value.
median 70
23
The Mode

• The mode is the data value that occurs most often
  in a data set.
• For example, the mode for the following set of
  numbers 7, 2, 4, 7, 8, 7, 10 turns out to be 7,
  because the number 7 occurs three times, more
  than any other number.

24
The Midrange

• The midrange is found by adding the lowest and
  highest data values and dividing the sum by 2.

25
The Midrange
midrange (2595)/2 120/2 60
26
Thinking Mathematically

• Section 4
• Measures of Dispersion

27
Dispersion

• The mean and the median were measures of central
  tendency they talk about the typical value
  who's in the middle.
• We also want to know about the other values. Are
  there a lot of values much higher than the mean?
  Much lower?
• That's called dispersion.

28
The Range

• The range, the difference between the highest and
  lowest data values in a data set, indicates the
  total spread of the data.
• Range highest data value - lowest data value
• For example, the ten most expensive markets for
  new homes in the U.S. has the following mean home
  cost in thousands of dollars 332, 256, 251,
  235, 223, 215, 215, 213, 210, 210.
• The range in costs is 332 - 210 122. (In
  other words 122,000).

29
Deviation

• The range tells us about the highest guy and the
  lowest guy. But what about the others?
• The deviation is the distance (positive or
  negative) between a value and the mean.
• If the mean is 62, a value of 98 is a distance of
  36 from the mean and a value of 55 is a distance
  of -7 from the mean.
• The deviation of a set of data is the sum of the
  deviations.

30
Deviation
95 54 66 88 63 35 40 95 90 67
32 -9 3 25 0 -28 -23 32 27 4
so the deviation is 0
630
0
10 values, so the mean is 63
31
Deviation

• Unfortunately, the deviation will always work out
  to be 0.
• It's totally useless
• For that and other reasons, dispersion is
  computed by what is called the standard deviation
  

32
Computing the Standard Deviation for a Data Set

• Find the mean of the data items.
• Find the deviation of each data item from the
  mean data item - mean
• Square each deviation (data item - mean)2
• Sum the squared deviations add up all of the
  (data item - mean)2

33
Computing the Standard Deviation for a Data Set

• Divide the sum in step 4 by n-1, where n
  represents the number of data item.
  
• Take the square root of the quotient in step 5.
  This value is the standard deviation for the data
  set. Standard deviation

34
Standard Deviation
95 54 66 88 35 63 40 95 90 67
32 -9 3 25 -28 0 -23 32 27 4
1024 81 9 625 784 0 529 1024 729 16
4821/9 535.67
the square root of 535.65 is 23.14
standard deviation 23.14
630
4821
10 values, so the mean is 63
35
Standard Deviation

• Therefore we can say that the average value was
  around 63, with typical values falling around 23
  points above or below that average.

36
Thinking Mathematically

• Section 5
• The Normal Distribution

37
Normal Distribution

• Whenever there's a lot of data centering around
  an average, the data is distributed in such a way
  that most values are around the average (the
  mean)
• This type of distribution is called a normal
  distribution or bell-shaped curve

38
Normal Distribution
99.7
95
68
- 3
- 2
-1
2
3
1
39
The 68-95-99.7 Rule for the Normal Distribution

• Approximately 68 of the measurements will fall
  within 1 standard deviation of the mean.
• Approximately 95 of the measurements will fall
  within 2 standard deviations of the mean.
• Approximately 99.7 (essentially all) the
  measurements will fall within 3 standard
  deviations of the mean.

40
The 68-95-99.7 Rule for the Normal Distribution
99.7
95
68
- 3
- 2
-1
2
3
1
41
Normal Distribution

• The best way to solve a "Normal Distribution
  Problem is
• Find the values of the mean and standard
  deviation
• Fill those numbers into the normal distribution
  graph
• See what range of values you're interested in
• Highlight that region of the graph
• Examine and find out the corresponding percentage.

42
Normal Distribution

• On the 1992 SAT's, the mean value on the
  quantitative portion was 510, with a standard
  deviation of 90.
• What percentage of the students scored between
  510 and 690?

43
The 68-95-99.7 Rule for the Normal Distribution

• mean 510, standard deviation of 90.

• percentage between 510 and 690?

99.7
95
47.5
68
- 3
510
600
420
690
330
780
240
- 2
-1
2
3
1
44
Normal Distribution

• On the 1992 SAT's, the mean value on the
  quantitative portion was 510, with a standard
  deviation of 90.
• What percentage of the students scored between
  510 and 690?
• 690 is 2 standard deviations above the mean
• 95 scored within 2 standard deviations
• half of those were above the mean
• 47.5

45
Computing z-Scores

• A z-score describes how many standard deviations
  a data item in a normal distribution lies above
  or below the mean. The z-score can be obtained
  using
• Data items above the mean have positive z-scores.
  Data items below the mean have negative z-scores.
  The z-score for the mean is 0.

46
Computing z-Scores

• In the previous example, a student scored 645.
  What was his z-score?
• z-score 135/90 1.5

47
Percentiles

• If n of the items in a distribution are less
  than a particular data item, we say that the data
  item is in the nth percentile of the
  distribution.
• For example, if a student scored in the 93rd
  percentile on the SAT, the student did better
  than about 93 of all those who took the exam.

48
Finding the Percentage of Data Items between Two
Given Items in a Normal Distribution

• Convert each given data item to a z-score
• Use the table to find the percentile
  corresponding to each z-score in step 1.
• Subtract the lesser percentile from the greater
  percentile and attach a sign.

49
Margin of Error in a Survey

• If a statistic is obtained from a random sample
  of size n, there is a 95 probability that it
  lies within of the true population statistic,
  where is called the margin of error.

50
Thinking Mathematically

• Section 6
• Scatter Plots, Correlation, and Regression Lines

51
Correlation

• Often we are not so much interested in
  characterizing one set of data, but trying to
  determine a relationship between two sets of
  data.
• For example, people were surveyed and asked
  three questions what is your gender? what is
  your age? how strongly do you believe in your
  religion?

52
Correlation

• For example, people were surveyed and asked three
  questions what is your gender? what is your
  age? how strongly do you believe in your
  religion?
• Is there a relationship between gender and
  religious practice? between age and religious
  practice?
• The strength of the relationship is called
  correlation.

53
Correlation

• One way to analyze correlation is by drawing a
  scatter plot and determinining from it the
  correlation coefficient.
• A scatter plot measures two pieces of data
  against each other.
• Let's take a look

54
Correlation Coefficient
r.7 moderate positive
r.85 strong positive
r1 perfect positive
55
Correlation Coefficient
r -.5 moderate negative
r -.85 strong negative
r -1 perfect negative
56




















Computing the Correlation Coefficient by Hand
n?xy - (?x)(?y)
r
n(?x2) - (?x)2 n(?y2) - (?y)2

• The formula is used to calculate the correlation
  coefficient, r.
• You have to be joking! This is MATH1101 !!!

57
Writing the equation of the Regression Line
by Hand

• The equation of the regression line is
• y mxb
• where

58
Values for Determining Correlationsin a
Population
?? .01
n
?? .05
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30
35 40 45 50 60 70 80 90 100
.999 .959 .917 .875 .834 .798 .765 .735 .708 .684
.661 .641 .623 .606 .590 .575 .561 .505 .463 .430
.402 .378 .361 .330 .305 .286 .269 .256
.950 .878 .811 .754 .707 .666 .632 .602 .576 .553
.532 .514 .497 .482 .468 .456 .444 .396 .361 .335
.312 .294 .279 .254 .236 .220 .207 .196