Summary Statistics: Measures of Location and Dispersion

Summary Statistics: Measures of Location and

Dispersion

nxxx 21

n

iix

1

The sum of values,

, can be denoted as

.

Select 4 students and ask “how many brothers and sisters do you have?”

•Data: 2, 3, 1, 3

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1

i

ix

9 xOr we can write

xccx

ncc ncxcxcx

Solve the following:

x4

3x

34x

234x

Measure of Central Tendency -Description of Average (Typical Value)

Sample Mean: nx

x

number of siblings – Data: 2, 3, 1, 3

Suppose we had selected a 5th person for our sample which had 10 siblings.

•New Data: 2, 3, 1, 3, 10

The sample mean is sensitive to extreme values and does not have to be a possible data value.

rank data from smallest to largest

if n is odd, median is the middle score

if n is even, median is the mean of two middle scores

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number of siblings – Data: 2, 3, 1, 3

New Data: 2, 3, 1, 3, 10

Sample median is not sensitive to extreme scores

Half the data will fall above the sample median and half below the sample median

  The median is a better measure of

central tendency if extreme scores exist.

If extreme scores are unlikely, the mean varies less from sample to sample than the median and is a better measure.

If the distribution is right skewed

If the distribution is symmetric

If the distribution is left skewed

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sample mode: most frequent score

Example: number of siblings – Data: 2,3,1,3Mode = 3

New Data: 2,3,1,3,10Mode = 3

Mode does not always exist/can be more than one

Also, it is unstable  Should be used with qualitative data

Example: number of siblings – Data: 2,3,1,3

Midrange =

New Data: 2,3,1,3,10

Midrange =

Midrange is totally dependent on extreme scores.

2

HighLow

22

31

2

HighLow

5.52

101

2

HighLow

Percentiles – gives the percentage below an observation

Quartiles – divide the data into four equally sized parts

  , First Quartile: 25th percentile

, Second Quartile ( ), 50th percentile

, Third Quartile, 75th percentile

1Q

2Q

3Q

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Order the data from smallest to largest

Find . This is

is the median of the lower half of the data; that is, it is the median of the data falling below (not including )

is the median of the upper half of the data; (same as above)

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1Q

2Q

2Q

2Q

3Q

Interquartile range (IQR) = Q3 – Q1

Range of the middle 50% of the data  5 number summary – The low score, Q1,

Q2, Q3, and the high score

Students Faculty0 0013555678 0 1 0 1 0552 2 045883 3 14 4 35 5 6 6 7 7 3

Students FacultyLow = 0 Low = 10Q1 = 1 Q1 = 15Q2 = 5 Q2 = 25Q3 = 7 Q3 = 31High = 10 High = 73

The box goes from Q1 to Q3 and represents IQR

The line through the box is Q2 ( )

Extreme values are identified by *’s

Lines, called whiskers, run from Q1 to the lowest value and from Q3 to the highest value (If the low or high are extreme then the whisker goes to the next value)

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Students Faculty

0

10

20

30

40

50

60

70

80

Stu

dent

s

CBA

43

38

33

A

Distribution #1 Distribution #21 1 52 5 2 553 5555555 3 5554 5 4 555 5 5 Distribution #1 Distribution #2 = 35 = 35 = 35 = 35mode = 35 mode = 35midrange =35 midrange = 35

X XX~

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Example: Years of experience of faculty Data: 1, 30, 22, 10, 5

   Range is sensitive to extreme scores

(Based entirely on the high and low)

Range is easy to compute

Large values of suggest large variability

It is difficult to interpret since it is in square units

Keep in mind it can never be negative

x

1111

X Squared of Sum222

2

nn

xxn

n

xx

n

SSX

nS

Example: Years of experience of faculty Data: 1, 30, 22, 10, 5

sample standard deviation – measures the average distance data points are from

Standard deviation is in the same units as the data

x

2SS

Z-score – Gives the number of standard deviations an observation is above or below the mean

Example: Test scores = 79, s = 9

If your score is 88%, what is your z-score?If your score is 63%, what is your z-score?

s

xxz

X

Approximately 68% of the data fall within 1 standard deviation of the mean

Approximately 95% of the data fall within 2 standard deviations of the mean

Approximately 99.7% of the data fall within 3 standard deviations of the mean

),( sxsx

)2,2( sxsx

)3,3( sxsx

Example: Suppose that the amount of liquid in

“12 oz.” Pepsi cans is a mound shaped distribution with oz. and s = 0.1 oz.

12x