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Prepared byMrs. Salwa Kamel

Revisoin

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Mean – Mode- Median

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Percentiles The pth percentile in an ordered array of n

values is the value in ith position, where

Example: Find the position of 60th percentile in an ordered array ( arrangement,) of 19 values?

It is the value in 12th position:

1)(n100

pi

121)(19100601)(n

100pi

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Quartiles Quartiles split the ranked data into 4 equal

groups 25% 25% 25% 25%

Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22

Example: Find the first quartile

(n = 9)

Q1 = 25th percentile, so find the (9+1) = 2.5 position

so use the value half way between the 2nd and 3rd values,

So Q1 = 12.5

25100

Q1 Q2 Q3

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Example

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Interquartile Range Can eliminate some outlier problems by using the

interquartile range

Eliminate some high-and low-valued observations and calculate the range from the remaining values.

Interquartile range = 3rd quartile – 1st quartile

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Comparing Coefficient of Variation

Stock A: Average price last year = $50 Standard deviation = $5

Stock B: Average price last year = $100 Standard deviation = $5

Both stocks have the same standard deviation, but stock B is less variable relative to its price

10%100%$50$5100%

xsCVA

5%100%$100

$5100%xsCVB

Best Measure of Center

Sample Standard Deviation Formula

(x – x)2

n – 1s =

Population Standard Deviation

2 (x – µ)

N =

This formula is similar to the previous formula, but instead, the population mean and population size are used.

Variance - Notation

s = sample standard deviation

s2 = sample variance

= population standard deviation

2 = population variance

the value midway between the maximum and minimum values in the original data set

Midrange

Midrange =maximum value + minimum value

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a. 5.40 1.10 0.42 0.73 0.48 1.10

b. 27 27 27 55 55 55 88 88 99

c. 1 2 3 6 7 8 9 10

Example Find Range and midrange for the following data

distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half.

Symmetric

(also called negatively skewed) have a longer left tail, mean and median are to the left of the mode

Skewed to the left

(also called positively skewed) have a longer right tail, mean and median are to the right of the mode

Skewed to the right

29 less 39 39 less 49 49 less 59 59 less 690123456789

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The Relative Positions of the Mean, Median and the Mode

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Measure of Skewness Describes the degree of departures of the

distribution of the data from symmetry. The degree of skewness is measured by the

coefficient of skewness, denoted as SK and computed as,

SDMedianMeanK

3S

a symmetric distribution has SK=0 since its mean is equal to its median and its mode.

Remark: a) If SK > 0, then the distribution is skewed to the right.b) SK < 0, then the distribution of the data set is skewed to left.c) If SK = 0, then the distribution is symmetric.

Example: Consider again the out – of – state tuition rates for the six school

sample from Pennsylvania. 4.9 6.3 7.7 8.9 7.7 10.3 11.7

1) Determine the following:1. Range2. Inter – quartile Range3. Standard Deviation4. Variance

2) Determine the direction of skewness of the preceding data.

Example

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Sample Population

x – µz =

Round z scores to 2 decimal places

Measures of Position z Score

z = x – xs

Interpreting Z Scores

Whenever a value is less than the mean, its corresponding z score is negative

Ordinary values: –2 ≤ z score ≤ 2

Unusual Values: z score < –2 or z score > 2

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Standard Normal Scores How many standard deviations away from the mean

are you?Standard Score (Z) =

“Z” is normal with mean 0 and standard deviation of 1.

Observation – meanStandard deviation

Z ScoreIt is a standard score that indicates how many SDs from the mean a particular values lies. Z = Score of value – mean of scores divided by standard deviation.

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Standard Normal Scores Example: Male Blood Pressure,

mean = 125, s = 14 mmHg

1) BP = 167 mmHg (Observation)

2) BP = 97 mmHg

3.014

125167Z

2.014

12597Z

Note that:

Ordinary values: –2 ≤ z score ≤ 2

Unusual Values: z score < –2 or z score > 2

 

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Standardizing Data: Z-Scores

X Z0 -1.34

2 -.80

4 -.27

6 .27

8 .80

10 1.34

Ordinary values: –2 ≤ z score ≤ 2

Unusual Values: z score < –2 or z score > 2

 

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Five Number Summary

2 3 8 8 9 10 10 12 15 18 22 63

Smallest LargestMedianQ1 Q3