test & measuement

V. SURESH KUMAR

Assistant Professor In Mathematics Rajalakshmi College Of Education Thoothukudi

TEST & MEASUREMENT RECORD – 2

STATISTICAL MEASURES

1. Statistics refers to numerical facts.2. Statistics as a science of collecting

summarizing, analyzing & interpreting numerical facts.

Statistics analysis:helps the teacher to describe or

summaries the test score which facilitates objectives comparison of student performance.

Statistical Analysis

Descriptive analysis Inferential analysis

Measure of Measure of Distributioncentral tendency variability

Mean Median Mode SD Skewness kurtosis

Inferential Analysis (General Linear Model)

T- test ANOVA ANCOVA Regression Analysis Distribution

Normal distribution Poison distribution

STEPS INVOLVED IN STATISTICAL ANALYSIS

(i). Collection of data. Collected by primary or secondary

method & tabulated in the numerical form. (ii). Classification of data.According to class intervals leads to

formation of frequency tables. (iii). Organisation & presentation. The presentation of data in the form

of class intervals & frequencies.

(iv). Selecting appropriate statistical technique

for analysis:Appropriate method of analysis

should be selected.(v). Applying selected method of analysis:

The computations are done & results

are obtained.(vi) Interpretation of results:

Results are interpreted & conclusions

are drawn.

ACHIEVEMENT TEST TOTAL MARK

91, 68, 85, 94, 82, 66, 58,98,80,76.

Marks in ascending order.

58,66,68,76,80,82,85,91,94,98

FREQUENCY DISTRIBUTION TABLE

Class interval

Real limit Tally Frequency

55 – 59 54.5 – 59.5 l 160 - 64 59.5 – 64.5 0 065 – 69 64.5 – 69.5 ll 270 – 74 69.5 – 74.5 0 075 – 79 74.5 – 79.5 l 1 80 – 84 79.5 – 84.5 ll 285 – 89 84.5 – 89.5 l 190 – 94 89.5 – 94.5 ll 295 – 99 94.5 – 99.5 l 1

MEASURES OF CENTRAL TENDENCY

1. Most of the items are gathering together or clustering around a particular point. This point is called “point of central tendency”.

2. The point is the most representative of the entire data.

Objectives:

(i). To get one single value that represents the entire data.

(ii). To facilitate comparison

Arithmetic Mean From Distributionadd all the values & divide the total by the total number of values.

sum of the itemM or X =

number of items x1 + x2 + x3 ……. Xn

= N x = N

58,66,68,76,80,82,85,91,94,98

sum of the items X =

number of items

= 58 +66+68+76+80+82+85+91+94+98

10 = 704

10 = 70.4

Arithmetic Mean From Frequency Distribution

1. Arithmetic mean by long method.2. Arithmetic mean by short method.

Arithmetic Mean by Long Method fxX =

NBy short method (Assumed Mean Method): fd X = A.M + x i N X - AM

d(deviation) = i A.M – Assumed Mean

fd X = A + × i NClass

intervalMid point

(X) f d = X – A / i fd

55 – 59 57 1 -4 -460 – 64 62 0 -3 065 - 69 67 2 -2 -470 – 74 72 0 -1 075 – 79 77 = A 1 0 080 – 84 82 2 1 2 85 – 89 87 1 2 290 – 94 92 2 3 695 – 99 97 1 4 4

6

Con/- 77- 57 - 20

d = = = - 4. 5 5 6

X = 77 + x 10 5 X = 89

MEDIAN (Md)Value which divides a distribution into

two parts.

middle most value when the given value

are arranged in an ascending or descending

order of magnitude.

b). When there is an even no/- of items:Average of the middle two scores is

taken as the median. sum of the middle two scores.Median = 2

Median for a series of ungrouped scores: N+1 thMedian = the measure in order of size. 2

MEDIAN FROM FREQUENCY DISTRIBUTION

N/2 - cfMedian = L + x i

f L – Exact lower limit of the C.I in which median lies (N/2th item lies).

cf – Cumulative frequency up to the lower limit of the CI containing median.

f – Frequency of the CI containing median.I – Size of class interval.

Median Real limit Frequency(f) Cumulative

frequency(cf) 54.5 – 59.5 1 159.5 – 64.5 0 164.5 – 69.5 2 369.5 – 74.5 0 374.5 – 79.5 1 4 = cf

L = 79.5 – 84.5 2 = f 684.5 – 89.5 1 789.5 – 94.5 2 9 94.5 – 99.5 1 10

Con/- N/2 = 10/2 = 5.

5 lie in the cumulative frequency 6.

therefore we have to select the total row itself.

cf = 4, f=2, L=79.5, i=5 5 - 4

Median = 79.5 + x 5 = 79.5 + 2.5

2 Median = 82

MODE Items occur in largest number of times.Mode for ungrouped data:Case (i)

Maximum number of repeated item referred as mode.Case (ii)

Adjacent scores have the same frequency and largest then mode is the sum of the two scores divided by two.

Case (iii)Non – adjacent values of items have the largest but equal frequency of two each, thus the mode is bimodal.

Mode

Mode = 3( median ) – 2( mean).

Mean = 89

Median = 82

Mode = 3(82) – 2(83)

= 246 – 166.

Mode = 80

Mode From Frequency Distribution f2 Mode = L+

f1 +f2

L - Lower limit of the mode class.

f1 - Frequency of the class interval preceding the mode class.

f2 – Frequency of the class interval succeed (above) the mode class.

i – size of the CI.

Mode = 3Median – 2 Mean.

It is called the crude mode or the empirical mode.

Measures Of Dispersion Or VariabilityIndividual items differ from their arithmetic mean

Objective of measuring variation:1. To test the reliability of an average.2. To serve as a basis for the control of

variability.3. To compute two or more groups with

regard to their variability.4. To facilitate the use of other statistical

measures.

Methods Of Studying Variation i). Range (R)

ii). Semi Inter quartile Range (or) quartile deviation.

iii). Mean deviation (M.D) or average deviation (A.D).

iv). Standard deviation(SD)

RANGE Difference between the highest & lowest

scores. R = H – L

Merit:1. When the data are too scattered to justify.2. Knowledge of total spread is wanted.3. Quick & crude estimate of variability.

Demerit:1. Not calculated for open end class intervals.2. Statistical analysis is difficult.3. It is unreliable when N is small.

Range

Range = Highest value – Lowest value.

= 98 – 58

= 40

The Semi – Inter – Quartile Range or Q

1. Extreme items is discarded, the limited range is more instructive.2. For this purpose, inter – quartile range – developed.

lowest lower upper upper quartile middle middle most quartile quartile quartile Q1 Q2 Q3

3. Half of the inter – quartile range or semi – inter – quartile range is called the quartile deviation (Q)

Q3 - Q1 Q = 2

N/4 – c.fQ1 = L1 + x i

f 3N/4 – c.f Q3 = L3 + x i f

Quartile Deviation Real limit Frequency Cumulative

frequency54.5 – 59.5 1 159.5 – 64.5 0 164.5 – 69.5 2 3 = cf

L= 69.5 – 74.5 0 = f 3 Q174.5 – 79.5 1 479.5 – 84.5 2 684.5 – 89.5 1 7 = cf

L= 89.5 – 94.5 2 = f 9 Q394.5 – 99.5 1 10

Q3 3N/4 = 3(10)/4 = 7.5L = 89.5, f= 2, cf = 7

7.5 - 7Q3 = 89.5 + x 5

4 = 89.5 + 1.25

Q3 = 90.75 Q1

N/4 = 10/4 = 2.5 L = 69.5, f = 0, cf = 3

2.5 – 3 Q1 = 69.5 + x 5

0Q1 = 69.5 +0

Q1 = 69.5 Q3 – Q1 Q = 2

90.75 – 69.5 Q = = 10.625

2 Q = 10.625

Average Deviation (AD) or Mean Deviation

Average distance between the mean & scores in the distribution.

X A.D =

N

X = X – M

= absolute value of deviation.

N – total No/- of scores.

Standard Deviation (S.D) or Rho 1. Introduced by karl pearson in 1893.2. Most reliable & stable index of variability.3. Greek letter sigma “ ”.4. A standard unit for measuring distances of

various score from their mean. In verbal term:

1. SD is the square root of the arithmetic mean of the squared deviations of measurement from their mean.

2. It is also called Root – Mean Square Deviation.

SD FROM UNGROUPED SCORES = X2 N SD from Grouped Scores:

fd2 fd 2

= i x - N N

Standard Deviation Class

interval Mid

point f d = X-A/i d²[

fd fxd²

55 - 59 57 1 -4 16 -4 1660 – 64 62 0 -3 9 0 065 - 69 67 2 -2 4 -4 870 – 74 72 0 -1 1 0 075 – 79 77 = A 1 0 0 0 080 – 84 82 2 1 1 2 285 – 89 87 1 2 4 2 490 – 94 92 2 3 9 6 18 95 – 99 97 1 4 16 4 16

6 66

66 6 2 = 5 -

10 10

= 5 x 6.24

= 5 x 2.49

= 12.45

CORRELATION A.M Tuttle

“ an analysis of the co-variation of two or more variables”.

Types of correlation:i. Positive correlation.ii. Negative correlation.

Positive correlation: Increase in one variable Increase in other variable.Decrease in one variable Decrease in other variable.Negative correlation:Decrease in one variable Increase in other variable.

Linear & Non – Linear Correlation 1. Variation in the values of two values of two

variables are in constant ratio.2. Y = a + bx – relationship.

Coefficient of correlation (r):1. To study the extent or degree of correlation

between two variable. 2. Represent by the letter r.

The range of r: 1. Correlation coefficient may assume values from 0 to 1 2. Minus (-) & plus(+) sign indicate the

relationship +ve or –ve.

r In Terms of Verbal Description

value of r verbal description ± 0.00 to ± 0.20 Independent or negligible relationship ±0.20 to ± 0.40 Low correlation present ±0.40 to ± 0.70 Substantial or marked ± 0.70 to ±1.00 High to very high

Computation of Correlation Coefficient1. Spearman’s Rank – difference method

.2. Pearson’s product moment method.

Spearman’s Rank Difference correlation coefficient ( (Rho)

6 D² ( = 1 - N (N²-1)

D – Difference in ranks.

Rank correlation Mark 1 Rank 1 Mark 2 Rank 2 D = R1-R2 D²

58 10 70 8 2 468 8 90 1 7 4966 9 71 7 2 476 7 89 2 5 2585 4 53 9 5 2582 5 76 5 0 080 6 49 10 4 1698 1 88 3 2 494 2 73 6 4 1691 3 86 4 1 1

144

Con/- 1 – (6 x144)

ρ = 10 (10 x10 -1) 1 - 864

ρ = = - 0.8717 990

Therefore ρ = - 0.87. his indicates a strong negative relationship between the ranks individuals obtained in the Math & English exam.

Graphical Representation of Data

Graph:1. Frequency distributions are converted into

visual models to facilitate understanding.2. Data may be presented through diagrams &

graphs.

Graphs of Frequency Distribution A frequency distribution can be presented graphically

in any of the following ways.1. Histogram.2. Frequency polygon. 3. Smoothed frequency curve. i. Greater than cumulative frequency

curve. ii. Less than cumulative frequency

curve. iii. Ogive.

Graphical representation of histogram

Graphical Representation of Frequency Polygon

Cumulative Frequency

Ogive curve