Chapter 4 Interpreting Histograms

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Chapter 4Interpreting Histograms

• Grouped Data are data presented in the form of a histogram, a frequency curve, an interval tally, or a similar display.

• An interval tally is a list of intervals and their frequency of scores.

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use class midpoint of classes for variable x

Mean from a Frequency Table

•m = class midpoint

• f = frequency

•N = total number

Estimated value of ScoresofNumberTotal

TotalScore

N

fmμ

k

iii

1

3

Standard Deviation from a Frequency Table

Estimated value of N

k

1i

2)(

ii mf

•m = class midpoint

•f = frequency

•N = total number=average

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Interval Frequency[32, 36) 3[28,32) 6[24,28) 12[20,24) 10 [16,20) 8[12,16) 2

ACT ScoresFind the average mean and standard deviation

Midpoint343026221814

Product10218031222014428

sum = 41 sum = 986

average mean is 986/41 = 24.04

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(mi - µ)9.965.961.96-2.04-6.04

-10.04

(mi - µ)2

99.2035.523.844.16

36.48100.8

fi(mi - µ)2

297.6213.1346.09941.616291.85201.16

sum = 1091.9

N

k

1i

2)(

ii mf

16.563.2690.1091

41

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Weighted Mean

x =w

(w • x)

A weighted mean of a group of scores is a mean computed in such a way that the frequency, or relative importance, of each score is taken into consideration

Used when the data values are assigned different weights, such as grades received and the computation of a GPA.

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Class Credit Grade

Math 4 B+

History 3 B

Physics 4 C

English 5 B-

Find the Grade Point Average

(4)(3.3)+3(3.0)+4(2.0)+5(2.6)

= 43.2

43.2/16 = 2.7

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Class Credit Grade

Starbursts 2 A

History 4 B+

Physics 3 A

Dieting 2 D-

Find Liz’s Grade Point Average

(2)(4.0)+4(3.33)+3(4.0)+2(.67) = 34.67

34.66/11 = 3.15

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x - s x x + s

68% within1 standard deviation

34% 34%

The Empirical Rule(applies to bell-shaped distributions)

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x - 2s x - s x x + 2sx + s


34% 34%

95% within 2 standard deviations


13.5% 13.5%

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x - 3s x - 2s x - s x x + 2s x + 3sx + s


34% 34%


99.7% of data are within 3 standard deviations of the mean


0.1% 0.1%2.4% 2.4%

13.5% 13.5%

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x


13

70 80 90 100 120 130110


34% 34%


99.7% of data are within 3 standard deviations of the mean

0.1% 0.1%2.4% 2.4%

13.5% 13.5%

IQ Scores have an average of 100 with a standard deviation of 10

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Chebyshev’s Theorem applies to distributions of any shape.

the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K2 , where K is any positive number greater than 1.

at least 3/4 (75%) of all values lie within 2 standard deviations of the mean.

at least 8/9 (89%) of all values lie within 3 standard deviations of the mean.

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ACT Scores


Product10218031222014428

average mean is 986/41 = 24.04 = 5.16

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What interval, centered around the mean in whichapproximately

a. 68% of the ACT scores?b. 95% of the ACT scores?c. 99% of the ACT scores?

x

2x

3x

a.

b.

c.

24.04 5.16

24.04 2(5.16)

24.04 3(5.16)

[18.9,29.2]

[13.72,34.36]

[8.58,39.44]

[8.58,36] Round to 36

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Using Chebyshev’s Theorem, what interval containsa. 3/4 of the ACT scores?b. 8/9 of the ACT scores?c. 15/16 of the ACT scores?

2x

3x

a.

b.

24.04 2(5.16)

24.04 3(5.16)

[13.74,34.12]

[8.58,36]

c. 4x 24.04 4(5.16) [3.12,36]

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Estimating the mode

Wba

aL

L lower limit of the modal interval W is the modal interval widtha and b are the differences in frequencies

ab

wL

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Estimating the median

Wf

F2N

L

L lower limit of the median interval W is the median interval widthN is the total number of scoresF is the sum of the frequencies up to but not including the median intervalf is the frequency of the median interval

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Wf

F2N

L

2 4 6 8 10

5

9

78

4

Find the estimated mode and median

Wba

aL

2

24

43

= 4.333

27

1233

5

4

= 5.714

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0 2 4 6 8 10

6

11

89

5

Find the estimated mean, standard deviation,mode and median. Find an interval that would contain 95% of the data.

39

59978511361 x

795.4x

554.2

]903.9,313.[)554.2(2795.42 x

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Wf

F2N

L

0 2 4 6 8 10

6

11

89

5

Find the estimated mean, standard deviation,mode and median. Find an interval that would contain 95% of the data.

Wba

aL

2

35

52

= 3.25

28

12

39

4

7

= 4.625

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Control Charts

A control chart is a graph that can be used to indicate how a series of new scores compare witha historical based mean and standard deviation

mean

mean + 1 standard deviation

mean - 1 standard deviation

Chapter 4 Interpreting Histograms

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