Unit 4 – Probability and Statistics

Section 7.7

Day 9

Warm-Up

P. 982 #5 - 12

Warm-Up Review

5) About 0.651

6) About 0.154

7) About 0.308

8) About 0.019

9) 0.5

10) About 0.265

11) About 0.505

12) About 0.145

Section 7.7 Statistics and Statistical GraphsGoal: Use measures of Central Tendency and Measures of

Dispersion to describe data sets, and use box-and whisker plots to describe data graphically.

Statistics – numerical values used to summarize and compare sets of data

2 Main GroupsMeasures of Central Tendency Measures of Dispersion (Variation)

Section 7.7 Statistics and Statistical Graphs

MEASURES OF CENTRAL TENDENCY Mean – the sum of data values divided by the

number of data values is a mean (average). Median – is the middle value of a data set. If

the data set contains a even number of values, the median is the mean of the two middle numbers

Mode – The most frequently occurring value in a set of data.

Example 1

Find the mean, median, and mode for the given data set.

36, 39, 40, 34, 48, 33, 25,

30, 37, 17, 42, 40, 24

Mean: Sum of the Terms Number of Terms

=44513

Mean: 34.2

Example 1 (cont.)

Find the mean, median, and mode for the given data set.

36, 39, 40, 34, 48, 33, 25,

30, 37, 17, 42, 40, 24

Median: Arrange terms from lowest to highest

17, 24, 25, 30, 33, 34, 36, 37, 39, 40, 40, 42, 48

Median: 36

Example 1 (cont.)

Find the mean, median, and mode for the given data set.

36, 39, 40, 34, 48, 33, 25,

30, 37, 17, 42, 40, 24

Mode: Number that appears the most

17, 24, 25, 30, 33, 34, 36, 37, 39, 40, 40, 42, 48

Mode: 40

Section 7.7 Statistics and Statistical Graphs

Box-and-Whisker Plot – a box and whisker plot uses quartiles to form the center box and whiskers.

Quartiles – separate a finite data set into four equal parts.

Outlier – is an item of data with a substantially different value from the rest of the items in the data set.

Quartiles

56 58 58 63 65 71 74 78 82 84 85 86

Median of data set Q2 = 72.5

Median of lower half Q1 = 60.5 Median of upper half Q3 = 83

71 58 56 63 84 74 85 82 86 78 65 58

71 + 742

= 72.5

58 + 632

= 60.5 82 + 842

= 83

Box-and-Whisker Plot

70 8060

Q2Q1 Q3

MaximumMinimum

56

50 90

60.5 72.5 83 86

Outlier

56 64 73 59 98 65 59

1 Find the mean, median, and mode of this data set.

2 Is there an outlier in this set.

3 If there is an outlier, remove it from the set and recalculate the mean, median, and mode.

67.71, 64, 59YES; 98

62.67, 61.5, 59

OutlierRules for outliers:

Maximum > 1.5(Median)

Minimum < ½(Median)

Given the data set:

22 40 42 45 50 58 64 73 65 65 83

Is there an outlier in this set.

Because: 22 < ½(58) 22 < 29

YES; 22

Measures of Dispersion (Variation)

Measure Definition

Range Greatest Value – Least Value

Interquartile Range

Q3 – Q1

Standard Deviation

Measure of how each data value in the set varies from the mean.

Measures of Variation

56 58 58 63 65 71 74 78 82 84 85 86

Median of data set Q2 = 72.5

Median of lower half Q1 = 60.5 Median of upper half Q3 = 83

1. What is the range for this data set?

2. What is the interquartile range for this data set?

30

22.5

How to find Standard Deviation

1. Find the mean of the data set.

2. Find the difference between each data value and the mean.

3. Square each difference.

4. Find the mean (average) of the squares.

5. Take the square root of the average. That is the standard deviation.

Data Set

56 58 58 63 65 71 74 78 82 84 85 86

Median of data set Q2 = 72.5

Median of lower half Q1 = 60.5 Median of upper half Q3 = 83

1. What is mean of this data set?

71.67

Standard Deviation Steps 2 & 3x Mean Difference Squared Value

56 71.67 -15.67 245.55

58 71.67 -13.67 186.87

58 71.67 -13.67 186.87

63 71.67 -8.67 74.17

65 71.67 -6.67 44.49

71 71.67 -0.67 0.45

74 71.67 2.33 5.43

78 71.67 6.33 40.07

82 71.67 10.33 106.71

84 71.67 12.33 152.03

85 71.67 13.33 177.69

86 71.67 14.33 205.35

SUM: 1425.28

Standard Deviation

Step 4: Find the mean of the squares.

Mean of the squares:1425.28

12

= 118.77

Step 5: Take square root of the mean of squares.

Sigma σ = sqrt(118.77)

σ = 10.9

10.9 is our Standard Deviation

HOMEWORK

P. 449

#4 – 7 ALL

#11 – 27 ODD