© asup-2007 frequency distribution, graph, and percentile 1

© aSup-2007 1

FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE

FREQUENCY DISTRIBUTION, GRAPH,

and PERCENTILE

© aSup-2007 2


A class of 40 students has just returned the Perceptual Speed test score. Aside from the primary question about your grade, you’d like to know more about how you stand in the class

How does your score compare with other in the class? What was the range of performance

What more can you learn by studying the scores?

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FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE Score of PERCEPTUAL SPEED

Test29 47 45 40 48 48 49 4540 49 48 37 48 46 55 6736 53 25 58 33 33 43 4232 51 48 54 47 40 38 4446 50 28 44 52 49 56 48

Taken from Guilford p.55

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FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE OVERVIE

W When a researcher finished the data collect

phase of an experiment, the result usually consist pages of numbers

The immediate problem for the researcher is to organize the scores into some comprehensible form so that any trend in the data can be seen easily and communicated to others

This is the jobs of descriptive statistics; to simplify the organization and presentation of data

One of the most common procedures for organizing a set of data is to place the scores in a FREQUENCY DISTRIBUTION

© aSup-2007 5


GROUPED SCORES After we obtain a set of measurement

(data), a common next step is to put them in a systematic order by grouping them in classes

With numerical data, combining individual scores often makes it easier to display the data and to grasp their meaning. This is especially true when there is a wide range of values.

© aSup-2007 6

FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE TWO GENERAL CUSTOMS IN

THE SIZE OF CLASS INTERVAL

1.We should prefer not fewer than 10 and more than 20 class interval.○ More commonly, the number class

interval used is 10 to 15.○ An advantage of a small number class

interval is that we have fewer frequencies which to deal with

○ An advantage of larger number class interval is higher accuracy of computation

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FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE TWO GENERAL CUSTOMS

IN THE SIZE OF CLASS INTERVAL

2. Determining the choice of class interval is that certain ranges of units (scores) are preferred.Those ranges are 2, 3, 5, 10, and 20.These five interval sizes will take care of almost all sets of data

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Test29 47 45 40 48 48 49 4540 49 48 37 48 46 55 6736 53 25 58 33 33 43 4232 51 48 54 47 40 38 4446 50 28 44 52 49 56 48


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HOW TO CONSTRUCT A GROUPED FREQUENCY DISTRIBUTION

Step 1 : find the lowest score and the highest score

Step 2 : find the range by subtracting the lowest score from the highest

Step 3 : divide the range by 10 and by 20 to determine the largest and the smallest acceptable interval widths. Choose a convenient width (i) within these limits

© aSup-2007 10


Score of PERCEPTUAL SPEED Test

29 47 45 40 48 48 49 45

40 49 48 37 48 46 55 67

36 53 25 58 33 33 43 42

32 51 48 54 47 40 38 44

46 50 28 44 52 49 56 48

Range = 42 42 : 10 = 4,2 and 42 : 20 = 2,1

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WHERE TO START CLASS INTERVAL

It’s natural to start the interval with their lowest scores at multiples of the size of the interval.

When the interval is 3, to start with 24, 27, 30, 33, etc.; when the interval is 4, to start with 24, 28, 32, 36, etc.

© aSup-2007 12


HOW TO CONSTRUCT A GROUPED FREQUENCY DISTRIBUTION

Step 4 : determine the score at which the lowest interval should begin. It should ordinarily be a multiple of the interval.

Step 5 : record the limits of all class interval, placing the interval containing the highest score value at the top. Make the intervals continuous and of the same width

Step 6 : using the tally system, enter the raw scores in the appropriate class intervals

Step 7 : convert each tally to a frequency

© aSup-2007 13


Test29 47 45 40 48 48 49 4540 49 48 37 48 46 55 6736 53 25 58 33 33 43 4232 51 48 54 47 40 38 4446 50 28 44 52 49 56 48


© aSup-2007 14


FREQUENCY DISTRIBUTION TABLESCORE

66 - 68

63 - 65

60 - 62

57 -59

54 - 56

51 - 53

48 - 50

45 - 47

42 - 44

39 - 41

36 - 38

33 - 35

30 - 32

27 - 29

24 - 26

SCORE

64 - 67

60 - 63

56 - 59

52 - 55

48 - 51

44 - 47

40 - 43

36 - 39

32 - 35

28 - 31

24 - 27

X max = 67

X min = 25

RANGE = 42Interval = 3

C.i = 15

Interval = 4

C.i = 11

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TALLYING THE FREQUENCIES Having adopted a set of class intervals,

we locate it within its proper interval and write a tally mark in the row for that interval.

Having completed the tallying, we count up the number case within each group to find the frequency (f), or the total number of case within the interval.

© aSup-2007 16


FREQUENCY DISTRIBUTION TABLESCORE TALLIES f

66 - 68

63 - 65

60 - 62

57 -59

54 - 56

51 - 53

48 - 50

45 - 47

42 - 44

39 - 41

36 - 38

33 - 35

30 - 32

27 - 29

24 - 26

SCORE TALLIES f

64 -67

60 - 63

56 - 59

52 - 55

48 - 51

44 - 47

40 - 43

36 - 39

32 - 35

28 - 31

24 - 27

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FREQUENCY DISTRIBUTION TABLESCORE TALLIES f

66 - 68 1

63 - 65 0

60 - 62 0

57 -59 1

54 - 56 3

51 - 53 3

48 - 50 10

45 - 47 6

42 - 44 4

39 - 41 3

36 - 38 3

33 - 35 2

30 - 32 1

27 - 29 2

24 - 26 1

SCORE TALLIES f

64 -67 1

60 - 63 0

56 - 59 2

52 - 55 4

48 - 51 11

44 - 47 8

40 - 43 5

36 - 39 3

32 - 35 3

28 - 31 2

24 - 27 1

© aSup-2007 18


SCORE LIMITS OF CLASS INTERVAL

The intervals are therefore labeled 24 to 27, 28 to 31, 32 to 35 and so on.

The top and bottom for each interval are called the score limit.

They are useful for labeling the intervals and in tallying score within the intervals.

© aSup-2007 19


EXACT LIMITS OF CLASS INTERVAL

In computations, however, it’s often necessary to work with exact limits.

A score of 40 actually means from 39.5 to 40.5 and that a score of 55 means from 54.5 to 55.5

Thus the interval containing scores 39 through 41 actually covers a range from 38.5 to 41.5

© aSup-2007 20


PERCEPTUAL

SPEED

SCORE f XcLower Exact

LimitUpper Exact

Limit

64 -67 1 65.5

60 - 63 0 61.5

56 - 59 2 57.5

52 - 55 4 53.5

48 - 51 11 49.5

44 - 47 8 45.5

40 - 43 5 41.5

36 - 39 3 37.5

32 - 35 3 33.5

28 - 31 2 29.5

24 - 27 1 25.5

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PERCEPTUAL

SPEED

SCORE f Xc Lower Exact Limit Upper Exact Limit

64 -67 1 65.5 63.5 67.5

60 - 63 0 61.5 59.5 63.5

56 - 59 2 57.5 55.5 59.5

52 - 55 4 53.5 51.5 55.5

48 - 51 11 49.5 47.5 51.5

44 - 47 8 45.5 43.5 47.5

40 - 43 5 41.5 39.5 43.5

36 - 39 3 37.5 35.5 39.5

32 - 35 3 33.5 31.5 35.5

28 - 31 2 29.5 27.5 31.5

24 - 27 1 25.5 23.5 27.5

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WARNING!! Although grouped frequency distribution can make

easier to interpret data, some information is lost. In the table, we can see that more people scored in

the interval 48 – 51 than in any other interval However, unless we have all the original scores to

look at, we would not know whether the 11 scores in this interval were all 48s, all, 49s, all 50s, or all 51 or were spread throughout the interval in some way

This problem is referred to as GROUPING ERROR The wider the class interval width, the greater the

potential for grouping error

© aSup-2007 23


STEM and LEAF DISPLAY In 1977, J.W. Tukey presented a

technique for organizing data that provides a simple alternative to a frequency distribution table or graph

This technique called a stem and leaf display, requires that each score be separated into two parts.

The first digit (or digits) is called the stem, and the last digit (or digits) is called the leaf.

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Data Stem & Leaf Display83 82 6362 93 7871 68 3376 52 9785 42 4632 57 5956 73 7474 81 76

3

4

5

6

7

8

9

3

2

1 6

5

2 3

2 66 2 7 9

4 3 8 4 6

2 8 3

2 1

3 7

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FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE GROUPED FREQUENCY DISTRIBUTION HISTOGRAM AND A STEM AND LEAF

DISPLAY

3 4 5 6 7 8 9

2 3

2 6

6 2

7

9

2 8

3

1 6

4

3

8 4

6

3 5

2

1

3 7

7654321

30 40 50 60 70 80 900

© aSup-2007 26


Place the following scores in a frequency distribution table2, 3, 1, 2, 5, 4, 5, 5, 1, 4, 2, 2

X f

5

4

3

2

1

3

2

1

4

2

LEARNING CHECK

© aSup-2007 27


A set of scores ranges from a high of X=142 to a low X=65○ Explain why it would not be reasonable to

display these scores in a regular frequency distribution table

○ Determine what interval width is most appropriate for a grouped frequency distribution for this set of scores

○ What range of values would form the bottom interval for the grouped table?

LEARNING CHECK

© aSup-2007 28


Initstereng!!Aoccdrnig to a rscheearch at an Elingsh uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is that frist and lsat ltteer is at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae we do not raed ervey lteter by it slef but the wrod as a wlohe.

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MAKING GRAPHPOLIGON and HISTOGRAM

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MAKING GRAPH

POLIGON

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PERCEPTUAL

SPEED

SCORE

f XcLower Exact

LimitLower Exact

Limit

64 -67 165.

563.5 67.5

60 - 63 061.

559.5 63.5

56 - 59 257.

555.5 59.5

52 - 55 453.

551.5 55.5

48 - 5111

49.5

47.5 51.5

44 - 47 845.

543.5 47.5

40 - 43 541.

539.5 43.5

36 - 39 337.

535.5 39.5

32 - 35 333.

531.5 35.5

28 - 31 229.

527.5 31.5

24 - 27 125.

523.5 27.5

© aSup-2007 32


POLIGON

X

f

0 29.5 37.5 45.5 53.5 61.525.5 33.5 41.5 49.5 57.5 65.5

12

10

8

6

4

2

21.5 69.5

Class Interval’s

MIDPOINT

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PERCEPTUAL SPEED

X

f

0 29.5 37.5 45.5 53.5 61.525.5 33.5 41.5 49.5 57.5 65.5

12

10

8

6

4

2

21.5 69.5

© aSup-2007 34


MAKING GRAPH

HISTOGRAM

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PERCEPTUAL

SPEED

SCORE

f XcLower Exact

LimitLower Exact

Limit

64 -67 165.

563.5 67.5

60 - 63 061.

559.5 63.5

56 - 59 257.

555.5 59.5

52 - 55 453.

551.5 55.5

48 - 5111

49.5

47.5 51.5

44 - 47 845.

543.5 47.5

40 - 43 541.

539.5 43.5

36 - 39 337.

535.5 39.5

32 - 35 333.

531.5 35.5

28 - 31 229.

527.5 31.5

24 - 27 125.

523.5 27.5

© aSup-2007 36


HISTOGRAM

X

f

0 27.5 35.5 43.5 51.5 59.5 67.523.5 31.5 39.5 47.5 55.5 63.5

12

10

8

6

4

2

Class Interval’s EXACT LIMIT

© aSup-2007 37


POLIGON and HISTOGRAM

X

f

0 27.5 35.5 43.5 51.5 59.5 67.523.5 31.5 39.5 47.5 55.5 63.5

12

10

8

6

4

2

© aSup-2007 38

FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE THE SHAPE OF A FREQUENCY

DISTRIBUTION

Symmetrical

It is possible to draw a vertical line through the

middle so that one side of the distribution is a mirror

image of the other

Skewed

The scores tend to pile up toward one end of the scale and taper off gradually at the other end

positive negative

© aSup-2007 39


Describe the shape of distribution for the data in the following table

X f

5

4

3

2

1

4

6

3

1

1

LEARNING CHECK

The distribution is negatively

skewed

© aSup-2007 40

FREQUENCY DISTRIBUTION, GRAPH, and PERCENTILE PERCENTILES and PERCENTILE

RANKS The percentile system is widely used in educational measurement to report the standing of an individual relative performance of known group. It is based on cumulative percentage distribution.

A percentile is a point on the measurement scale below which specified percentage of the cases in the distribution falls

The rank or percentile rank of a particular score is defined as the percentage of individuals in the distribution with scores at or below the particular value

When a score is identified by its percentile rank, the score called percentile

© aSup-2007 41


Suppose, for example that A have a score of X=78 on an exam and we know exactly 60% of the class had score of 78 or lower….…

Then A score X=78 has a percentile of 60%, and A score would be called the 60th percentile

Percentile Rank refers to a percentage

Percentile refers to a score

© aSup-2007 42


X f cf c%

5

4

3

2

1

1

5

8

4

2

CUMMULATIVE FREQUENCY and CUMULATIVE PERCENTAGE

20

19

14

6

2

100%

95%

70%

30%

10%

1.What is the 95th percentile?

Answer: X = 4.5

2.What is the percentile rank for X = 3.5

Answer: 70%

© aSup-2007 43


INTERPOLATION It is possible to determine some

percentiles and percentile ranks directly from a frequency distribution table

However, there are many values that do not appear directly in the table, and it is impossible to determine these values precisely

© aSup-2007 44


INTERPOLATIONUsing the following distribution of scores we will find the percentile rank corresponding to X=7

X f cf c%

1098765

284641

2523151151

100926044204

Notice that X=7 is located in the interval bounded by the real limits of 6.5 and 7.5

The cumulative percentage corresponding to these real limits are 20% and 44% respectively

© aSup-2007 45


Scores (X) – percentage

7.5 44%

7.0 …….. ??

6.5 20%

STEP 1

For the scores, the width of the interval is 1 point.

For the percentage, the width is 24 points

STEP 2

Our particular score is located 0.5 point from the top of the interval. This is exactly halfway down the interval

STEP 3

Halfway down on the percentage scale would be

½ (24 points) = 12 points

STEP 4For the percentage, the top of the interval is 44%, so 12 points down would be 32%

© aSup-2007 46


Using the following distribution of scores we will use interpolation to find the 50th percentile

X f cf c%

20 - 24

15 - 19

10 - 14

5 - 9

0 - 4

2

3

3

10

2

20

18

15

12

2

100

90

75

60

10

A percentage value of 50% is not given in the table; however, it is located between 10% and 60%, which are given.

These two percentage values are associated with the upper real limits of 4.5 and 9.5

© aSup-2007 47


Scores (X) – percentage

9.5 60%

?? …….. 50%

4.5 10%

STEP 1

For the scores, the width of the interval is 5 point.

For the percentage, the width is 50 points

STEP 2

The value of 50% is located 10 points from the top of the percentage interval.

As a fraction of the whole interval this is 1/5 of the total interval

STEP 3

Using this fraction, we obtain 1/5 (5 points) = 1 point

The location we want is 1 point down fom the top of the score interval

STEP 4Because the top of the interval is 9.5, the position we want is 9.5 – 1 = 8.5 the 50th percentile = 8.5

© aSup-2007 48


On a statistics exam, would you rather score at the 80th percentile or at the 40th percentile?

For the distribution of scores presented in the following table,

LEARNING CHECK

X f cf c%

40 - 4930 - 3920 - 2910 - 19

0 - 9

46

1032

25211552

1008460208

a.Find the 60th percentile

b.Find the percentile rank for X=39.5

c. Find the 40th percentile

d.Find the percentile rank for X=32

© aSup-2007 49


SCOREFrequen

cy

57 -59 1

54 – 56 3

51 – 53 4

48 – 50 8

45 – 47 9

42 – 44 7

39 – 41 6

36 – 38 5

33 – 35 3

30 – 32 2

27 – 29 1

24 – 26 1

H O M E W O R Ka.Make a polygon or histogram

graph for the distribution of scores presented in the following table

b.Describe the shape of distribution

c. Find the 25th, 50th, and 75th percentile

d.Find the percentile rank for X=25, X=50, and X=75

© aSup-2007 50


SCORE

Frequency

Xc

Exact Limit

Lower

Upper

57 - 59

158

56.5 59.5

54 – 56

355

53.5 56.5

51 – 53

452

50.5 53.5

48 – 50

849

47.5 50.5

45 – 47

946

44.5 47.5

42 – 44

743

41.5 44.5

39 – 41

640

38.5 41.5

36 – 38

537

35.5 38.5

33 – 35

334

32.5 35.5

30 – 32

231

29.5 32.5

27 – 29

128

26.5 29.5

24 – 26

125

23.5 26.5

e.Make a polygon or histogram graph for the distribution

Histogram

PolygonXc

E.L.

© asup-2007 frequency distribution, graph, and percentile 1

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