Sum = is the sign commonly used to indicate the algebraic sum (_i.e._ the difference between the sum of the minus quant.i.ties and the plus quant.i.ties).
_x y _= products of deviation in one trait multiplied by deviation in the other trait with appropriate sign.
Applying the formula we find: =================================================================== |ARITH-| | | SPEL- | | | | |METIC | x | x^2 | LING | y | y^2 | xy | --+------+---+------------+-------+---+-------------+-------------+ A | 1|-3 | 9| 2|-6 | 36| +18| B | 2|-2 | 4| 4|-4 | 16| +8| C | 3|-1 | 1| 6|-2 | 4| +2| D | 4| 0 | 0| 8| 0 | | | E | 5|+1 | 1| 10|+2 | 4| +2| F | 6|+2 | 4| 12|+4 | 16| +8| G | 7|+3 | 9| 14|+6 | 36| +18| | ___| | __| ___| | ___| __| | 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum xy = +56| |Av. =4| | |Av. =8 | | | | =================================================================== Sum x y +56 +56 r = ---------------------------- = --------------------- = ---- = +1 (sqrt(Sum x^2)(sqrt(Sum y^2) (sqrt(28))(sqrt(112)) 56
If instead of achievement in one field being positively related (going together) in the highest possible degree, these individuals show the opposite type of relationship, _i.e.,_ the maximum negative relationship (this might be expressed as opposition--a place above the average in one achievement going with a correspondingly great deviation below the average in the other achievement), then our coefficient becomes -1.
Applying the formula:
=================================================================== |ARITH-| | | SPEL- | | | | |METIC | x | x^2 | LING | y | y^2 | x*y | --+------+---+------------+-------+---+-------------+-------------+ A | 1|-3 | 9| 14|+6 | 36| -18| B | 2|-2 | 4| 12|+4 | 16| -8| C | 3|-1 | 2| 10|+2 | 4| -2| D | 4| 0 | | 8| 0 | | | E | 5|+1 | 2| 6|-2 | 4| -2| F | 6|+2 | 4| 4|-4 | 16| -8| G | 7|+3 | 9| 2|-6 | 36| -18| | ___| | __| ___| | ___| __| | 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum xy = -56| |Av. =4| | |Av. =8 | | | | ===================================================================
It will be observed that in this case each plus deviation in one achievement is accompanied by a minus deviation for the other trait; hence, all of the products of _x_ and _y_ are minus quant.i.ties. (A plus quant.i.ty multiplied by a plus quant.i.ty or a minus quant.i.ty multiplied by a minus quant.i.ty gives us a plus quant.i.ty as the product, while a plus quant.i.ty multiplied by a minus quant.i.ty gives us a minus quant.i.ty as the product.)
(Sum xy) -56 -56 r = ------------------------------ = ------------------- = ---- = -1.
(sqrt(Sum x^2))(sqrt(Sum y^2)) (sqrt(28)sqrt(112)) = 56
If there is no relationship indicated by the measures of achievements which we have found, then the coefficient of correlation becomes 0. A distribution of scores which suggests no relationship is as follows:
================================================================= |ARITH- | | | | | | |METIC | x | x^2 |Spelling | y | y^2 | x.y --+-------+----+-----------+---------+----+-------------+-------- | | | | | | | - + A | 2 | -2 | 4 | 12 | +4 | 16 | -8 +6 B | 1 | -3 | 9 | 8 | 0 | | 0 +4 C | 4 | 0 | | 2 | -6 | 36 | 0 +4 D | 5 | +1 | 1 | 14 | +6 | 36 | -6 E | 3 | -1 | 1 | 4 | -4 | 16 | -14 +14 F | 7 | +3 | 9 | 6 | -2 | 4 | G | 6 | +2 | 4 | 10 | +2 | 4 | | ____| | | ___ | | | | |28 | |Sum x^2=28 | 7|56 | | Sum y^2=112 | xy=0 | AV.=4 | | | AV.=8 | | | ===================================================================
(Sum xy) 0 r = ---------------------------- = ------------------- = 0.
(sqrt(Sum x^2)sqrt(Sum y^2)) (sqrt(28)sqrt(112))
In a similar manner, when the relationship is largely positive as would be indicated by a displacement of each score in the series by one step from the arrangement which gives a +1 coefficient, the coefficient will approach unity in value.
=============================================================== ARITHMETIC| x | x^2 |SPELLING| y | y^2 | ---+------+----+-----------+--------+----+------------+-------- A |1 | -3 |9 |4 | -4 | 16 |+ 12 B |2 | -2 |4 |2 | -6 | 36 |+ 12 C |3 | -1 |1 |8 | 0 | |+ 4 D |4 | 0 | |6 | -2 | 4 |+ 4 E |5 | +1 |1 |12 | +4 | 16 |+ 18 F |6 | +2 |4 |10 | +2 | 4 |Sxy=50 G |7 | +3 |9 |14 | +6 | 36 | |Av. =4| |Sum x^2 =28|Av. = 8 | |Sum y^2= 112| ===============================================================
Sum xy +50 r= -------------------------- = ---- = +.89.
sqrt(Sum x^2)sqrt(Sum y^2) 56
Other ill.u.s.trations might be given to show how the coefficient varies from + 1, the measure of the highest positive relationship (going together) through 0 to -1, the measure of the largest negative relationship (opposition). A relationship between traits which we measure as high as +.50 is to be thought of as quite significant. It is seldom that we get a positive relationship as large as +.50 when we correlate the achievements of children in school work. A relationship measured by a coefficient of .15 may _not_ be considered to indicate any considerable positive or negative relationship. The fact that relationships among the achievements of children in school subjects vary from +.20 to +.60 is a clear indication of the fact that abilities of children are variable, or, in other words, achievement in one subject does not carry with it an _exactly corresponding_ great or little achievement in another subject. That there is some positive relationship, _i.e.,_ that able pupils tend on the whole to show all-round ability and the less able or weak in one subject _tend_ to show similar lack of strength in other subjects, is also indicated by these positive coefficients.
QUESTIONS
1. Calculate the median point in the following distribution of eighth-grade composition scores on the Hillegas scale.
Quality 0 18 26 37 47 58 67 Frequency 2 68 73 3
2. Calculate the median point in the following distribution of third-grade scores on the Woody subtraction scale.
No. problems 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Frequency 2 2 2 3 3 5 4 5 8 16 16 16 23 20 21 11 22 11 2
22 23 24 + 1
3. Compare statistically the achievements of the children in two eighth-grade cla.s.ses whose scores on the Courtis addition tests were as follows:
Cla.s.s A--6, 5, 8, 9, 7, 10, 13, 4, 8, 7, 8, 7, 6, 8, 15, 6, 7, 0, 6, 9, 5, 8, 7, 10, 8, 4, 7, 8, 6, 9, 5, 7, 2, 6, 8, 5, 7, 8, 7, 8, 5, 8, 10, 6, 3, 6, 8, 17, 5, 7.
Cla.s.s B--10, 4, 8, 13, 11, 9, 8, 10, 7, 9, 11, 10, 18, 7, 12, 9, 10, 8, 11, 10, 12, 9, 2, 11, 8, 10, 9, 14, 11, 7, 10, 12, 10, 6, 11, 8, 10, 9, 10, 17, 8, 11, 9, 7, 9, 11, 8, 12, 9, 13.
4. If the marks received in algebra and in geometry by a group of high school pupils were as given below, what relationship is indicated by the coefficient of correlation?
|GEOMETRY |ALGEBRA |MARKS |MARKS 1. |80 |60 2. |68 |73 3. |65 |80 4. |96 |80 5. |59 |62 6. |75 |65 7. |90 |75 8. |86 |90 9. |52 |63 10. |70 |55 11. |63 |54 12. |85 |95 13. |93 |90 14. |87 |70 15. |82 |68 16. |79 |75 17. |78 |86 18. |79 |75 19. |82 |60 20. |70 |82 21. |52 |86 22. |94 |85 23. |72 |73 24. |53 |62 25. |94 |85
5. Compare the abilities of the 10-year-old pupils in the sixth grade with the abilities of the 14-year-old pupils in the same grade, in so far as these abilities are measured by the completion of incomplete sentences.
(Note: 5 = 5.0-5.999.)
================================================== NO. SENTENCES | | COMPLETED | 10-YEAR-OLDS | 14-YEAR-OLDS --------------+--------------+-------------------- 24 |-- |-- 23 |-- |-- 22 |-- |-- 21 |1 |-- 20 |-- |-- 19 |-- |-- 18 |-- |-- 17 |-- |1 16 |3 |-- 15 |-- |2 14 |7 |4 13 |10 |3 12 |18 |7 11 |9 |10 10 |7 |9 9 |8 |10 8 |2 |10 7 |3 |10 6 |-- |2 5 |2 |3 4 |-- |2 3 |-- |-- 2 |-- |1 1 |-- |-- 0 |-- |-- ===========================================
6. From the scores given here, calculate the relationship between ability to spell and ability to multiply. Use the average as the central tendency.
============================== PUPIL|SPELLING|MULTIPLICATION -----+--------+--------------- A |9 |22 B |10 |16 C |2 |19 D |6 |14 E |13 |24 F |8 |22 G |10 |17 H |7 |20 I |3 |21 J |2 |21 K |14 |20 L |8 |18 M |7 |23 N |11 |25 O |8 |25 P |17 |24 Q |10 |21 R |4 |16 S |9 |15 T |6 |19 U |12 |22 V |14 |19 W |8 |17 X |3 |20 Y |11 |18 ==============================