Kinds of Statistics

Purpose Target Characteristics

DescriptiveStatistics Summarizing,

Describing Sample Statistic

Inferential Statistics

Analyzing, Generalization Population Parameter

Tabulation of Data

Ungrouped Ungrouped Data Data

Grouped Data Grouped Data

151599

161613131111101088

1212

1616151513131212111110109988

Frequency Distribution

• Graphic description of how many times a score or group of scores occurs in a sample

• Common symbol is “f”

Absolute Frequency

Score(X) Score(X) Frequency(f) Frequency(f) 1616151513131212111110109988

1111225544441122

Frequency distribution

Score (X) Score (X) Absolute Absolute FrequencyFrequency

Relative Relative FrequencyFrequency

1616151513131212111110109988

1111225544441122

0.050.050.050.050.100.100.250.250.200.200.200.200.050.050.100.10

Σx = 94 N= 20N= 20 1.001.00

Frequency distributionScore Score

(X) (X) Absolute Absolute

FrequencyFrequencyRelative Relative

FrequencyFrequency PercentagePercentage

1616151513131212111110109988

1111225544441122

0.050.050.050.050.100.100.250.250.200.200.200.200.050.050.100.10

0.05x100=50.05x100=50.05x100=50.05x100=50.10x100=10.10x100=1000.25x100=20.25x100=2550.20x100=20.20x100=2000.20x100=20.20x100=2000.05x100=50.05x100=50.10x100=10.10x100=100

Σx = 94 N= 20N= 20 1.001.00 100100

Cumulative Frequency

• Cumulative frequency distribution is a graphic depiction of the how many times groups of scores appear in a sample

• Common symbol is “cf”

• “cf “ is used to compute percentile scores

Cumulative frequency

XX ff cfcf

1616

1515

1313

1212

1111

1010

99

8 8

11

11

22

55

44

44

11

2 2

2020

1919

1818

1616

1111

77

33

2 2

Percentile score showing relative standing in a distribution showing what percentage of scores are higher and lower than

a certain score.Percentile computation cf

P(percentile) = (100) ----- N

N= number of scores Cf =cumulative frequency Cf 16 P= (100)----- = (100)------ = 80 N 20 Cf 20 P = (100)-----= (100) ------ = 100 N 20

Bar graph

Frequency Polygon

Measures of Central Tendency

• Mean: arithmetic average of all scores in a distribution

• Median: the point at which exactly half of the scores in a distribution are below & half are above

• Mode: most frequently occurring score(s)

Measures of central tendency1. Mean / arithmetic average _ Σx X = ------- Σx = sum of all scores

N N = number of scores

Example:Example:13 + 14 + 15 + 16 + 17 = 7513 + 14 + 15 + 16 + 17 = 75

ΣΣx = 75x = 75 N= 5N= 5

_ _ ΣΣx 75x 75 X = -------X = ------- = = -------= 15-------= 15 NN 5 5

2. ModeXX f f

1616

1515

131312

1111

1010

99

88

11

11

225

44

44

11

2 2

1. Odd number 13, 15, 16, 17, 19

2. Even number

12, 13, 12, 13, 15, 1715, 17, 18, 19, 18, 19 = = 1616

3.Median:

Measures of Variability

• These describe the spread, or dispersion, of scores in a distribution

• These measures describe the nature & extent to which scores vary

• Three most commonly used measures are:

1. Range2. Variance

3. Standard Deviation

Measures of variability1. Range

13, 15, 16, 17, 19 19-13= 6

2.2. Variance (V)Variance (V)

V =------------- N -1

3. standard deviation3. standard deviation

XX (X - )(X - ) ΣΣ(X - )(X - )22

19191818171716161515141413 13

+ 3+ 3+2+2+1+100-1-1-2-2-3 -3

9944110011449 9

112112 00 2828

ExampleExample

Σx2 28 (V( = ---------= --------=4.6 N-1 7-1

S = 2.14

Normal /bell-shaped curve

Properties of Normal /bell-shaped curve

• It is a symmetrical distribution• Most of the scores tend to occur near the center– while more extreme scores on either side of the center

become increasingly rare. – As the distance from the center increases, the

frequency of scores decreases. • The mean, median, and mode are the same.

Normal Probability Curve

• Describes an expected distribution of scores in a population or sample

• More than 2/3 of scores cluster in the middle of the curve

• Scores that are extremely high or low are sometimes called outliers

• Shaped like a bell

Normal /bell-shaped curve

Symmetrical vs. asymmetrical d distribution

• In a symmetrical distribution– the part of the histogram on the left side of the fold

would be the mirror image of the part on the right side of the fold.

• In a asymmetrical, distribution• the two sides will not be mirror images of each other.

True symmetric distributions include what we will later call the normal distribution.

• A asymmetric distribution is either Positively or negatively skewed. – In a positively skewed distribution the scores

cluster toward the lower end of the scale (that is, the smaller numbers) with increasingly fewer scores at the upper end of the scale (that is, the larger numbers).

– With a negatively skewed distribution, most of the scores occur toward the upper end of the scale while increasingly fewer scores occur toward the lower end.

Kurtosis Distribution• Mesokurtic distribution : with normal

distribution of scores• Leptokurtic distribution: packed, with low

variability of scores • Platykurtic Distribution: flat, with high

variability of scores

Examples based on the curveAdults intelligence = 100 SD= 15

Mean +1 SD = 68%100 + 15 = 85 -115

34 percent = 100--115 34 percent = 85 -- 100

Mean + 2 SD = 94%100 + (2 X 15) = 70 --130

115 + 15 = 13013 % = 115 – 130

85 – 15 = 7013 % = 70 –85

Standard ScoresTo compare scores on different measurement scales

I. Z-Scores: the commonest score

Z-score propertiesHow many scores above/below the meanThe mean being set at zeroThe SD being set at one

Standard Scores

II. T-Score: A standard score whose distribution has a mean of 50 and a standard deviation of 10.

Advantages of T-scoreEnabling us to work with whole numbersAvoiding describing subjects’ performances

with negative numbers

An example:

_

25- 20 = ---------- = +1 5

X - 30-35 Z = ---------- = -------- = -1 SD 5Ali’s Z-score : +1

Better than 84% of the classAhmad’s Z-score: --1

Better than only 16% 0f the class

Student X SD

Ali 25 5 20

Ahmad 30 5 35

An example:

_

X – 76- 54 Z = --------- = ---------- = +1.1 SD 20

X - 82- 72 Z = ---------- = -------- = 0.67 SD 15So, using standard scores, Ali did better than Ahmad because

Ali’s mark was more standard deviations above the class mean than Ahmad’s score was above his own class mean.

Student X SD

Ali 76 20 54

Ahmad 82 15 72

Negative SkewNegative Skew Positive skewPositive skew

Test items were easy.

Testees performed well.

The score are far from zero.

Test items were difficult.

Testees performed poorly.

The scores are near zero.

Skewed Distribution

39

The Coefficient of Correlation, r

The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. It requires interval or ratio-scaled data.

It can range from -1.00 to 1.00.Values of -1.00 or 1.00 indicate perfect and

strong correlation.Values close to 0.0 indicate weak correlation.Negative values indicate an inverse relationship

and positive values indicate a direct relationship.

40

Perfect Correlation

41

Correlation Coefficient - Interpretation

CorrelationGo-togetherness of variablesGo-togetherness of variablesNo cause-effect relationshipNo cause-effect relationshipBetween -1 and +1Between -1 and +1

Types of correlation:1.Pearson Product-moment

a. Used for interval data

43

Correlation Coefficient - Formula

2. Spearman rank-order, rhoa. used for ranked or ordinal data

StudentsStudents Teacher 1Teacher 1 Teacher 2Teacher 2 DD DD22

AA

BB

CC

DD

E E

11

22

33

44

5 5

55

44

33

22

1 1

44

22

00

22

4 4

1616

44

00

44

16 16

55 1515 1515 1212 4040

6 (40) 240 P = 1-- ––––––– = ––––––––

5 (25-1) 120

1 – 2 = -1

Standard error of measurement (SEM)True score= Raw score – Random errors

How to estimate

X = scoren = number of items

An example:X= 79 n = 100

X (n- X) 79 (100-79) SEMX = –––––––– = –––––––––––= 4

n – 1 100-1

79 + 1SEM = 75-83 68% 79 + 2SEM = 71-87 95%

Point-biserial correlation:

rpbi = point-biserial correlation coefficient

Mp = whole-test mean for students answering item correctly (i.e., those coded as 1s)

Mq = whole-test mean for students answering item incorrectly (i.e., those coded as 0s)

St = standard deviation for whole test

p = proportion of students answering correctly (i.e., those coded as 1s)

q = proportion of students answering incorrectly (i.e., those coded as 0s)

As another example, where

the whole-test mean for Ss answering correctly is

30;

the whole-test mean for Ss answering incorrectly is

45;

the standard deviation for the whole test is still

8.29;

the proportion of Ss answering correctly is still .50;

and the proportion answering incorrectly is

still .50.

Types of Correlation coefficientsType of Correlation

CoefficientTypes of Scales

Pearson product-moment

Both scales interval (or ratio)

Spearman rank-order

Both scales ordinal

Phi Both scales are naturally dichotomous (nominal)

BiserialOne scale artificially dichotomous

(nominal), one scale interval (or ratio)

Point-biserial One scale naturally dichotomous (nominal), one scale interval (or ratio)

Gamma One scale nominal, one scale ordinal

Correction for guessing

A student has taken a test of 100 items. As he has no knowledge, he takes choice A and marks it for all the items. His true / corrected score is

75 Corrected score= 25--

------------ 4—1

25-25= 0

Some other factors affecting a person’s score:

Practice effectCoaching effectCeiling effectTest compromiseTest MethodTest Taker’s characteristics

In a normal distribution, what percentage of scores fall between the mean and one standard deviation? a. 35% b. 50%

c. 68% d. 95%

A test has been given to 100 students. Twenty students have obtained the score of 50. What is the percentage of this score?

a. 10 b. 15 c. 20 d. 30

f 20P x= ------ X 100 = ------= 0.20 X 100 = 20 N 100

A test is administered to 100 students. The cumulative frequency of the score of 50 is 40. How many students have scores below the score of 50?

a. about 25 b. about 20 c. about 40 d. about 30

N=100 cf=40 cf 40 P= (100)----- = 100 x ------ = 40 N 100

In a test eight of the students obtained a score of 85. This score has the highest frequency. What is the label used for this score?a. Mean b. Mode c. Median d. Range

Fry’s Readability Graph

Directions for Use Randomly select three 100-word passages from a book or an

article. Plot the average number of syllables and the average number of

sentences per 100 words on the graph to determine the grade level of the material.

Choose more passages per book if great variability is observed and conclude that the book has uneven readability.

Few books will fall into the solid black area, but when they do, grade level scores are invalid.

Additional Directions• Randomly select three sample passages and count exactly 100

words beginning with the beginning of a sentence. Don't count numbers. Do count proper nouns.

• Count the number of sentences in the hundred words, estimating length of the fraction of the last sentence to the nearest 1/10th.

• Count the total number of syllables in the 100-word passage. • Enter graph with average sentence length and number of

syllables; plot dot where the two lines intersect. Area where dot is plotted will give you the approximate grade level.

• If a great deal of variability is found, putting more sample counts into the average is desirable.