Lesson03

IBS Statistics Year 1

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What we are going to learn?

• Review

• Chapter 3: Dispersion• Range• Variance (SD2)• Standard Deviation (SD)• Coefficient of variation (CV)

• Chapter 4: Displaying and exploring data• Dotplot• Stem-leaf• Boxplot• Skewness

ReviewDiscrete counting or Continuous measuring

• Class size

• Age

Review

Chapter 3: Dispersion–Range–Variance (SD2)–Standard Deviation (SD)

–Coefficient of variation (CV)

Chapter 4: Displaying and exploring data–Dotplot–Stem-leaf–Boxplot–Skewness • Temperature

• Sales volume

• Salary

• Height

• Weight

• Shoe size (NL)

Review



Chapter 4: Displaying and exploring data–Dotplot–Stem-leaf–Boxplot–Skewness

Review

P46. N.30 Ch.2

25 = 32, 26 = 64, suggests 6 classes

i = 88.33> 571- 416

Use interval of 100

Constructing Frequency Distribution: Quantitative Data

45 observations

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P46. N.30 Ch.2

0relative

Class interval =

100

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P87 N.60 Ch.3

SCCoast, an Internet provider in the Southeast, developed the following frequency distribution on the age of Internet users. Describe the central tendency:

X = 2410 / 60 = 40.17 (years)

Central Tendency : Mean, Mode, Median

Mean: Average Median: Midpoint Mode: Most Frequency

Mode = 45 (years) Median = ? (years)

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Lm=(60+1)/2=30.5 Value:40 50

Location: 28 48

30.5

30.5-2848-28 =

M-4050-40

Median= 41.25

Step 1: Define the location of the median Step 2: Calculate the median

P87 N.60 Ch.3

M

Chapter 3 Dispersion

Dispersion

Range

Interquartile Range

Variance (SD2) and Standard Deviation (SD)

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Coefficient of variation (CV)

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Dispersion: – tells us about the spread of the data. – Help us to compare the spread in two or more

distributions.

Mean is not reliable

Chapter 3 Dispersion

RangeRange:is the difference between the largest and the smallest

value in a data set.

Example:To find the range in 3,5,7,3,11

Range = 11-3 = 8

Review




Variance

Population Variance:• is the mean of the squared difference between each

value and the mean. • overcomes the weakness of the range by using all the

values in the population.

Sample Variance:

Nμ)-Σ(X

=σ2

2

1-n)X-Σ(X

=s2

2

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Population Variance:N

μ)-Σ(X=σ

22

27

EXAMPLE – Variance and Standard Deviation

The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance?

Step 1: Get the mean

Step 2: Find the difference between each observation and the mean

Step 3: Square the difference and sum up Step 4: Divided by N

Variance

Population Standard Deviation:is the square root of the population variance.

Sample Standard Deviation:is the square root of the sample variance.

2σ=σ

2s=s

Standard DeviationReview




Example:The hourly wages earned by a sample of five students are:

€7, €5, €11, €8, €6. Find the variance and standard deviation.

Step 1: Get the meanStep 1: Get the mean

Step 2: Sum up the squared differences

Step 2: Sum up the squared differences

Step 3: Divided by N-1Step 3: Divided by N-1

Step 4: Square root itStep 4: Square root it

7.40=537

=nΣX

=X

( ) ( ) ( )

5.30=1-5

21.2=

1-57.4-6+...+7.4-7

=1-nX-XΣ

=s222

2

s = €2.30

The variance is €5.30; the standard deviation is €2.30.

Standard DeviationReview




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Schiphol Utrecht20 40 50 60 80 20 49 50 51 80

Compare

• The number of coffee sales in Utrecht Starbucks is more closely clustered around the mean of 50 than for the sales number in Schiphol

Starbucks.

Standard Deviation

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Standard Deviation of Grouped Data

Step 2: Use f * (M-Xmean)2

Step 3: Sum up

P87 N.60 Ch.3

Step 1: Find the Midpoint

Step 4: Divided by N-1 709860-1

Step 5: Square root it 709860-1

= 10.97

Review




Coefficient of VariationThis is the ratio of the standard deviation to the mean:

The coefficient of variation describes the magnitude sample values and the variation within them.

The following times were recorded by the quarter-mile and mile runners of a university

track team (times are in minutes).

Quarter-Mile Times: 0.92 0.98 1.040.90 0.99

Mile Times: 4.52 4.35 4.60 4.704.50

After viewing this sample of running times, one of the coaches commented that the quarter milers turned in the more consistent times. Calculate the appropriate measure to check this and comment on the coach’s statement.We can compare the dispersion with the coefficient of variation because they have different “magnitudes”.

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The following times were recorded by the quarter-mile and mile runners of a university

track team (times are in minutes).

Quarter-Mile Times: 0.92 0.98 1.040.90 0.99

Mile Times: 4.52 4.35 4.60 4.704.50

After viewing this sample of running times, one of the coaches commented that the quarter milers turned in the more consistent times. Calculate the appropriate measure to check this and comment on the coach’s statement.We can compare the dispersion with the coefficient of variation because they have different “magnitudes”.

Coefficient of variation of Q-Mile Times is: 0.05639/0.966=0.05837==>6%Coefficient of variation of Mile Times is: 0.12954/4.534=0.02857==>3%No, the mile-time team showed more consistent times.

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Chapter 4 Displaying and Exploring Data

Dot plots:

Stem-and-Leaf Displays:Each numerical value is divided into two parts. The leading

digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis.

Leaf

Stem

Chapter 4 Displaying and Exploring DataReview




Stem-and-Leaf Displays:





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Quartiles, Deciles, and PercentilesAlternative ways of describing spread of data include determining thelocation of values that divide a set of observations into equal parts.



Quartiles, Deciles, and Percentiles

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95 1 25 100

93 1 24 96

88 2 23 92

85 3 21 84

79 1 18 72

75 4 17 68

70 6 13 52

65 2 7 28

62 1 5 20

58 1 4 16

54 2 3 12

50 1 1 4

N = 25

Raw PercentileScore Frequency Frequency Rank

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43 61 9175101 104

Example:


The first quartile is ?

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Organize the data from lowest to largest valueStep 1:Step 1:

43 61 9175101 104

L25 = (n+1) = (6+1) =1.75 Step 2:Step 2: P100

25100

P1 P2 P3 P4 P5 P6

Draw two linesStep 3:Step 3:

43 6161-43 = 18

P1.75

P1 P20.75



Draw two linesStep 3:Step 3:

43 6161-43 = 18

P1 P20.75 * 18 = 13.5

43+13.5 = 56.5

The first quartile is 56.5.






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Listed below, ordered from smallest to largest, are the number of visits last week.

The median is 58.

a. Determine the median number of calls.

Q1 = 51.25 Q3 = 66.00

b. Determine the first and third quartiles.

P110. N.14 Ch.4

Exercise

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Listed below, ordered from smallest to largest, are the number of visits last week.

D1 = 45.30 D9 = 76.40

c. Determine the first decile and the ninth decile.

P33 = 53.53

d. Determine the 33rd percentile.

P110. N.14 Ch.4

Exercise

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Box Plots

A graphical display, based on quartiles to visualize a set of data.


minimumminimum Q1Q1 MedianMedian Q3Q3 maximummaximum

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Box Plots



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Box Plots & Cumulative Frequency Distribution





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skewed

Skewness:

Another characteristic of a set of data is the shape.

• symmetric, • positively skewed, • negatively skewed, • bimodal.





Zero skewness

mode=median=mean

Zero skewness

mode=median=mean

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positive skewness

Mode median mean

positive skewness

Mode median mean

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negative skewness

Mode median mean

negative skewness

Mode median mean

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Skewness: • symmetric, • positively skewed, • negatively skewed, • bimodal.


ExerciseA sample of 28 time shares in the Orlando, Florida, area revealed the following daily charges for a one-bedroom suite. For convenience the data are ordered from smallest to largest. Construct a box plot to represent the data. Comment on the distribution. Be sure to identify the first and third quartiles and the median.

• The median is $253.

• About 25% of the semi-private rooms are less than $214 and 25% above $304.

• The distribution is negatively skewed.

P113. N.18 Ch.4

Review




• Review

• Chapter 3: Dispersion• Range• Variance (SD2)• Standard Deviation (SD)• Coefficient of variation (CV)

• Chapter 4: Displaying and exploring data• Dotplot• Stem-leaf• Boxplot• Skewness

What we have learnt today?

Lesson03

Education

review chapter

standard deviation sd

coefficient of variation

ibs statisticsyear

variance sd2