Chapter X: Part 10

His companions during his college course and even before that
time were great enfluences. The surroundings of De Quincy were
enfluences. Not only De Quincy's habit of opium but other
habits which were peculiar to his life.

His marriage to the woman which he did not especially care
for.

The many well educated and noteworthy friends of De Quincy.

VALUE 585. Written by a boy in the fourth year of the high
school, aged 16 years

_Fluellen_

The passages given show the following characteristic of
Fluellen: his inclination to brag, his professed knowledge of
History, his complaining character, his great patriotism,
pride of his leader, admired honesty, revengeful, love of fun
and punishment of those who deserve it.

VALUE 675. Written by a girl in the first year of the high
school, aged 18 years

_Ichabod Crane_

Ichabod Crane was a schoolmaster in a place called Sleepy
Hollow. He was tall and slim with broad shoulders, long arms
that dangled far below his coat sleeves. His feet looked as if
they might easily have been used for shovels. His nose was
long and his entire frame was most loosely hung to-gether.

VALUE 772. Written by a boy in the third year of the high
school, aged 16 years

_Going Down with Victory_

As we road down Lombard Street, we saw flags waving from
nearly every window. I surely felt proud that day to be the
driver of the gaily decorated coach. Again and again we were
cheered as we drove slowly to the postmasters, to await the
coming of his majestie's mail. There wasn't one of the gaily
bedecked coaches that could have compared with ours, in my
estimation. So with waving flags and fluttering hearts we
waited for the coming of the mail and the expected tidings of
victory.

When at last it did arrive the postmaster began to quickly
sort the bundles, we waited anxiously. Immediately upon
receiving our bundles, I lashed the horses and they responded
with a jump. Out into the country we drove at reckless
speed--everywhere spreading like wildfire the news, "Victory!"
The exileration that we all felt was shared with the horses.
Up and down grade and over bridges, we drove at breakneck
speed and spreading the news at every hamlet with that one cry
"Victory!" When at last we were back home again, it was with
the hope that we should have another ride some day with
"Victory."

VALUE 838. Written by a boy in the Freshman class in college

_Venus of Melos_

In looking at this statue we think, not of wisdom, or power,
or force, but just of beauty. She stands resting the weight of
her body on one foot, and advancing the other (left) with knee
bent. The posture causes the figure to sway slightly to one
side, describing a fine curved line. The lower limbs are
draped but the upper part of the body is uncovered. (The
unfortunate loss of the statue's arms prevents a positive
knowledge of its original attitude.) The eyes are partly
closed, having something of a dreamy langour. The nose is
perfectly cut, the mouth and chin are moulded in adorable
curves. Yet to say that every feature is of faultless
perfection is but cold praise. No analysis can convey the
sense of her peerless beauty.

VALUE 937. Written by a boy in the Freshman class in college

_A Foreigner's Tribute to Joan of Arc_

Joan of Arc, worn out by the suffering that was thrust upon
her, nevertheless appeared with a brave mien before the Bishop
of Beauvais. She knew, had always known that she must die when
her mission was fulfilled and death held no terrors for her.
To all the bishop's questions she answered firmly and without
hesitation. The bishop failed to confuse her and at last
condemned her to death for heresy, bidding her recant if she
would live. She refused and was lead to prison, from there to
death.

While the flames were writhing around her she bade the old
bishop who stood by her to move away or he would be injured.
Her last thought was of others and De Quincy says, that recant
was no more in her mind than on her lips. She died as she
lived, with a prayer on her lips and listening to the voices
that had whispered to her so often.

The heroism of Joan of Arc was wonderful. We do not know what
form her great patriotism took or how far it really led her.
She spoke of hearing voices and of seeing visions. We only
know that she resolved to save her country, knowing though she
did so, it would cost her her life. Yet she never hesitated.
She was uneducated save for the lessons taught her by nature.
Yet she led armies and crowned the dauphin, king of France.
She was only a girl, yet she could silence a great bishop by
words that came from her heart and from her faith. She was
only a woman, yet she could die as bravely as any martyr who
had gone before.

The following compositions have been evaluated by Professor Thorndike, and may be used to supplement the scale given above.

VALUE 13

Last Monday the house on the corner of Jay street was burned
down to the ground and right down by Mrs. brons house there is
a little child all alone and there is a bad man sleeping in
the seller, but we have a wise old monkey in the coal ben so
the parents are thankful that they don't have to pay any
reward.

VALUE 20

Some of the house burned and the children were in bed and
there were four children and the lady next store broke the
door in and went up stars and woke the peple up and whent out
of the house when they moved and and the girl was skard to
look out of the window and all the time thouhth that she saw a
flame.

And the wise monkey reward from going to the firehouse and
jumping all round and was thankful from his reward and was
thankful for what he got. $15. was his reward.

VALUE 30

A long time ago, I do not know, how long but a man and a woman
and a little boy lived together also a monkey a pet for the
little boy it happened that the man and the woman were out,
and the monkey and little boy, and the house started to burn,
and the monkey took the little boys hand, and, went out.

The father had come home and was glad that the monkey had
saved his little boy.

And that, monkey got a reward.

VALUE 40

Once upon a time a woman went into a dark room and lit a
match. She dropped it on the floor and it of course set the
house afire.

She jumped out of the window and called her husband to come
out too.

They both forgot all about the baby. All of a sudden he
appeared in the window calling his mother.

His father had gone next door to tel afone to the fire house.

They had a monkey in the house at the time and he heard the
child calling his mother.

He had a plan to save the baby.

He ran to the window where he was standing. He put his tail
about his waist and jumped off the window sill with the baby
in his tail.

When the people were settled again they gave him a silver
collar as a reward.

VALUE 50

A University out west, I cannot remember the name, is noted
for its hazing, and this is what the story is about. It is the
hazing of a freshman. There was a freshman there who had been
acting as if he didn't respect his upper class men so they
decided to teach him a lesson. The student brought before the
Black Avenger's which is a society in all college to keep the
freshman under there rules so they desided to take him to the
rail-rode track and tie him to the rails about two hours
before a train was suspected and leave him there for about an
hour, which was a hour before the 9.20 train was expected. The
date came that they planned this hazing for so the captured
the fellow blindfolded him and lead him to the rail rode
tracks, where they tied him.

VALUE 60

I should like to see a picture, illustrating a part of
L'allegro. Where the godesses of Mirth and Liberty trip along
hand in hand. Two beautiful girls dressed in flowing garments,
dancing along a flower-strewn path, through a pretty garden.
Their hair flowing down in long curls. Their countenances
showing their perfect freedom and happiness. Their arms
extended gracefully smelling some sweet flower. In my mind
this would make a beautiful picture.

VALUE 70

It was between the dark and the daylight when far away could
be seen the treacherous wolves skulking over the hills. We sat
beside our campfires and watched them for awhile. Sometimes a
few of them would howl as if they wanted to get in our camp.
Then, half discouraged, they would walk away and soon there
would be others doing the same thing. They were afraid to come
near because of the fires, which were burning brightly. I
noticed that they howled more between the dark and the
daylight than at any time of the night.

VALUE 80

The sun was setting, giving a rosy glow to all the trees
standing tall black against the faintly tinted sky. Blue,
pink, green, yellow, like a conglomeration of paints dropped
carelessly onto a pale blue background. The trees were in such
great number that they looked like a mass of black crepe, each
with its individual, graceful form in view. The lake lay
smooth and unruffled, dimly reflecting the beautiful coloring
of the sky. The wind started madly up and blew over the lake's
glassy surface making mysterious murmurings blending in with
the chirping songs of the birds blew through the tree tops
setting the leaves rustling and whispering to one another. A
squirrel ran from his perch chattering, to the lofty
branches--a far and distant hoot echoed in the silence, and
soon night, over all came stealing, blotting out the scenery
and wrapping all in restful, mysterious darkness.

VALUE 90

Oh that I had never heard of Niagara till I beheld it! Blessed
were the wanderers of old, who heard its deep roar, sounding
through the woods, as the summons to an unknown wonder, and
approached its awful brink, in all the freshness of native
feeling. Had its own mysterious voice been the first to warn
me of its existence, then, indeed, I might have knelt down and
worshipped. But I had come thither, haunted with a vision of
foam and fury, and dizzy cliffs, and an ocean tumbling down
out of the sky--a scene, in short, which nature had too much
good taste and calm simplicity to realize. My mind had
struggled to adapt these false conceptions to the reality, and
finding the effort vain, a wretched sense of disappointment
weighed me down. I climbed the precipice, and threw myself on
the earth feeling that I was unworthy to look at the Great
Falls, and careless about beholding them again.

A scale for measuring English composition in the eighth grade, which takes account of different types of composition, such as narration, description, and the like, has been developed by Dr. Frank W. Ballou, of Boston.[27] For those interested in the following up of the problem of English composition this scale will prove interesting and valuable.

Several scales have been developed for the measurement of the ability of children in reading. Among them may be mentioned the scale derived by Professor Thorndike for measuring the understanding of sentences.[28] This scale calls attention to that element in reading which is possibly the most important of them all, that is, the attempt to get meanings. We are all of us, for the most part, concerned not primarily with giving expression through oral reading, but, rather, in getting ideas from the printed page. A sample of this scale is given on the following page.

* * * * *

SCALE ALPHA. FOR MEASURING THE UNDERSTANDING OF SENTENCES

Write your name here...............................
Write your age.............years............months.

SET _a_

Read this and then write the answers. Read it again as often as you need to.

John had two brothers who were both tall. Their names were Will and Fred. John's sister, who was short, was named Mary. John liked Fred better than either of the others. All of these children except Will had red hair. He had brown hair.

1. Was John's sister tall or short?.....................
2. How many brothers had John?..........................
3. What was his sister's name?..........................

SET _b_

Read this and then write the answers. Read it again as often as you need to.

Long after the sun had set, Tom was still waiting for Jim and Dick to come. "If they do not come before nine o'clock," he said to himself, "I will go on to Boston alone." At half past eight they came bringing two other boys with them. Tom was very glad to see them and gave each of them one of the apples he had kept. They ate these and he ate one too. Then all went on down the road.

1. When did Jim and Dick come?...................................
2. What did they do after eating the apples?.....................
3. Who else came besides Jim and Dick?...........................
4. How long did Tom say he would wait for them?..................
5. What happened after the boys ate the apples?..................

SET _c_

Read this and then write the answers. Read it again as often as you need to.

It may seem at first thought that every boy and girl who goes to school ought to do all the work that the teacher wishes done. But sometimes other duties prevent even the best boy or girl from doing so. If a boy's or girl's father died and he had to work afternoons and evenings to earn money to help his mother, such might be the case. A good girl might let her lessons go undone in order to help her mother by taking care of the baby.

1. What are some conditions that might make even the best boy leave
school work unfinished?............................................
...................................................................
2. What might a boy do in the evenings to help his family?.........
3. How could a girl be of use to her mother?.......................
4. Look at these words: _idle, tribe, inch, it, ice, ivy, tide, true,
tip, top, tit,
tat, toe._

Cross out every one of them that has an _i_ and has not any _t_ (T) in it.

SET _d_

Read this and then write the answers. Read it again as often as you need to.

It may seem at first thought that every boy and girl who goes to school ought to do all the work that the teacher wishes done. But sometimes other duties prevent even the best boy or girl from doing so. If a boy's or girl's father died and he had to work afternoons and evenings to earn money to help his mother, such might be the case. A good girl might let her lessons go undone in order to help her mother by taking care of the baby.

1. What is it that might seem at first thought to be true, but really is
false?
.......................................................................

2. What might be the effect of his father's death upon the way a boy
spent
his
time?.................................................................
3. Who is mentioned in the paragraph as the person who desires to have
all lessons completely
done?..............................................
.......................................................................

4. In these two lines draw a line under every 5 that comes just after a
2,
unless the 2 comes just after a 9. If that is the case, draw a line
under
the next figure after the 5:

5 3 6 2 5 4 1 7 4 2 5 7 6 5 4 9 2 5 3 8 6 1 2 5 4 7 3 5 2 3 9 2 5 8 4 7
9 2 5 6
1 2 5 7 4 8 5 6

* * * * *

Many tests have been devised which have been thought to have more general application than those which have been mentioned above for the particular subjects. One of the most valuable of these tests, called technically a completion test, is that derived by Dr. M.R. Trabue.[29] In these tests the pupil is asked to supply words which are omitted from the printed sentences. It is really a test of his ability to complete the thought when only part of it is given. Dr. Trabue calls his scales language scales. It has been found, however, that ability of this sort is closely related to many of the traits which we consider desirable in school children. It would therefore be valuable, provided always that children have some ability in reading, to test them on the language scale as one of the means of differentiating among those who have more or less ability. The scores which may be expected from different grades appear in Dr. Trabue's monograph. Three separate scales follow.

* * * * *

_Write only one word on each blank_
_Time Limit: Seven minutes_ NAME ..........................

TRABUE
LANGUAGE SCALE B

1. We like good boys................girls.
6. The................is barking at the cat.
8. The stars and the................will shine tonight.
22. Time................often more valuable................money.
23. The poor baby................as if it.....................sick.
31. She................if she will.
35. Brothers and sisters ................ always ................ to
help..............other and should................quarrel.
38. ................ weather usually................ a good effect
................ one's spirits.
48. It is very annoying to................................tooth-ache,
................often comes at the most................time
imaginable.
54. To................friends is always................the........
it takes.

_Write only one word on each blank_
_Time Limit: Seven minutes_ NAME..........................

TRABUE
LANGUAGE SCALE D

4. We are going................school.
76. I................to school each day.
11. The................plays................her dolls all day.
21. The rude child does not................many friends.
63. Hard................makes................tired.
27. It is good to hear................voice.......................
..........friend.
71. The happiest and................contented man is the one........
........lives a busy and useful.................
42. The best advice................usually................obtained
................one's parents.
51.................things are................ satisfying to an ordinary
................than congenial friends.
84.................a rule one................association..........
friends.

_Write only one word on each blank_
_Time Limit: Five minutes_ NAME ............................

TRABUE
LANGUAGE SCALE J

20. Boys and................soon become................and women.
61. The................are often more contented.............. the
rich.
64. The rose is a favorite................ because of................
fragrance and.................
41. It is very................ to become................acquainted
................persons who................timid.
93. Extremely old..................sometimes..................almost as
.................. care as ...................
87. One's................in life................upon so............
factors ................ it is not ................ to state any
single................for................ failure.
89. The future................of the stars and the facts of............
history are................now once for all,................I
like them................not.

* * * * *

Other standard tests and scales of measurement have been derived and are being developed. The examples given above will, however, suffice to make clear the distinction between the ordinary type of examination and the more careful study of the achievements of children which may be accomplished by using these measuring sticks. It is important for any one who would attempt to apply these tests to know something of the technique of recording results.

In the first place, the measurement of a group is not expressed satisfactorily by giving the average score or rate of achievement of the class. It is true that this is one measure, but it is not one which tells enough, and it is not the one which is most significant for the teacher. It is important whenever we measure children to get as clear a view as we can of the whole situation. For this purpose we want not primarily to know what the average performance is, but, rather, how many children there are at each level of achievement. In arithmetic, for example, we want to know how many there are who can do none of the Courtis problems in addition, or how many there are who can do the first six on the Woody test, how many can do seven, eight, and so on. In penmanship we want to know how many children there are who write quality eight, or nine, or ten, or sixteen, or seventeen, as the case may be. The work of the teacher can never be accomplished economically except as he gives more attention to those who are less proficient, and provides more and harder work for those who are capable, or else relieves the able members of the class from further work in the field. It will be well, therefore, to prepare, for the sake of comparing grades within the same school or school system, or for the sake of preparing the work of a class at two different times during the year, a table which shows just how many children there are in the group who have reached each level of achievement. Such tables for work in composition for a class at two different times, six months apart, appear as follows:

DISTRIBUTION OF COMPOSITION SCORES FOR A SEVENTH GRADE

======================================
| NUMBER OF CHILDREN
+-----------------------
| NOVEMBER | FEBRUARY
--------------+-----------+-----------
Rated at 0 | 0 | 0
1.83 | 1 | 1
2.60 | 6 | 4
3.69 | 12 | 6
4.74 | 8 | 11
5.85 | 3 | 4
6.75 | 1 | 3
7.72 | 1 | 2
8.38 | 0 | 1
9.37 | 0 | 0
======================================

A study of such a distribution would show not only that the average performance of the class has been raised, but also that those in the lower levels have, in considerable measure, been brought up; that is, that the teacher has been working with those who showed less ability, and not simply pushing ahead a few who had more than ordinary capacity. It would be possible to increase the average performance by working wholly with the upper half of the class while neglecting those who showed less ability. From a complete distribution, as has been given above, it has become evident that this has not been the method of the teacher. He has sought apparently to do everything that he could to improve the quality of work upon the part of all of the children in the class.

It is very interesting to note, when such complete distributions are given, how the achievement of children in various classes overlaps. For example, the distribution of the number of examples on the Courtis tests, correctly finished in a given time by pupils in the seventh grades, makes it clear that there are children in the fifth grade who do better than many in the eighth.

THE DISTRIBUTION OF THE NUMBER OF EXAMPLES CORRECTLY FINISHED
IN THE GIVEN TIME BY PUPILS IN THE SEVERAL GRADES

=================================================================== ADDITION | SUBTRACTION No. OF |----------------------+ No. OF |------------------------ EXAMPLES| GRADES | EXAMPLES | GRADES FINISHED| 5 | 6 | 7 | 8 | FINISHED | 5 | 6 | 7 | 8 --------+----+-----+-----+-----+----------+----+-----+-----+------- 0 | 12 | 15 | 5 | 4 | 0 | 6 | 2 | 2 | -- 1 | 26 | 23 | 14 | 9 | 1 | 5 | 6 | 2 | 1 2 | 27 | 31 | 8 | 6 | 2 | 7 | 8 | 1 | -- 3 | 31 | 27 | 27 | 9 | 3 | 13 | 21 | 3 | 1 4 | 25 | 28 | 19 | 16 | 4 | 21 | 18 | 13 | 2 5 | 16 | 23 | 16 | 15 | 5 | 26 | 30 | 12 | 7 6 | 15 | 22 | 12 | 12 | 6 | 17 | 27 | 15 | 9 7 | 1 | 11 | 8 | 9 | 7 | 15 | 27 | 18 | 9 8 | 3 | 4 | 6 | 11 | 8 | 15 | 20 | 12 | 12 9 | 1 | 2 | 3 | 8 | 9 | 10 | 13 | 9 | 12 10 | -- | -- | -- | 6 | 10 | 8 | 6 | 13 | 11 11 | -- | -- | 1 | -- | 11 | 6 | 2 | 3 | 12 12 | -- | -- | 1 | 2 | 12 | 3 | 1 | 7 | 9 13 | -- | -- | -- | -- | 13 | 2 | 2 | 3 | 5 14 | -- | -- | -- | -- | 14 | 1 | 1 | 3 | 7 15 | -- | -- | -- | 2 | 15 | -- | -- | 2 | 3 16 | -- | -- | -- | 1 | 16 | -- | -- | 1 | 2 17 | -- | -- | -- | -- | 17 | -- | 1 | -- | 1 18 | -- | -- | -- | -- | 18 | -- | -- | -- | 1 19 | -- | -- | -- | -- | 19 | -- | -- | -- | 4 20 | -- | -- | -- | -- | 20 | -- | -- | -- | 2 21 | -- | -- | -- | -- | 21 | -- | -- | -- | 1 22 | -- | -- | -- | -- | 22 | -- | -- | -- | -- --------+----+-----+-----+-----+----------+----+-----+-----+------- Total | | | | | | | | | papers |157 | 86 | 119 | 111 | |155 | 185 | 119 | 111 ===================================================================

THE DISTRIBUTION OF THE NUMBER OF EXAMPLES CORRECTLY FINISHED
IN THE GIVEN TIME BY PUPILS IN THE SEVERAL GRADES

======================================================================= MULTIPLICATION | DIVISION ------------------------------------|---------------------------------- No. of | GRADES |No. of | GRADES Examples|---------------------------|Examples|------------------------- Finished| 5 | 6 | 7 | 8 |Finished| 5 | 6 | 7 | 8 --------|------+-----+-----+--------|--------|------+-----+-----+------ 0 . . .| 10 | 4 | -- | -- | 0 . . .| 17 | 7 | 1 | -- 1 . . .| 10 | 4 | 3 | -- | 1 . . .| 19 | 17 | 2 | 1 2 . . .| 19 | 20 | 5 | 1 | 2 . . .| 18 | 22 | 8 | 4 3 . . .| 21 | 17 | 11 | 5 | 3 . . .| 21 | 26 | 6 | 2 4 . . .| 28 | 31 | 16 | 3 | 4 . . .| 25 | 27 | 8 | 6 5 . . .| 26 | 34 | 12 | 13 | 5 . . .| 21 | 27 | 11 | 7 6 . . .| 24 | 27 | 13 | 13 | 6 . . .| 9 | 15 | 12 | 4 7 . . .| 9 | 20 | 16 | 10 | 7 . . .| 10 | 15 | 16 | 18 8 . . .| 5 | 14 | 21 | 19 | 8 . . .| 6 | 7 | 20 | 9 9 . . .| 3 | 9 | 11 | 13 | 9 . . .| 4 | 7 | 11 | 6 10 . . .| -- | 4 | 6 | 10 |10 . . .| 4 | 9 | 7 | 13 11 . . .| 1 | -- | 2 | 9 |11 . . .| 1 | 3 | 3 | 7 12 . . .| -- | -- | 2 | 6 |12 . . .| -- | 2 | 10 | 10 13 . . .| -- | -- | 1 | 3 |13 . . .| -- | 2 | -- | 10 14 . . .| -- | -- | -- | 3 |14 . . .| 1 | -- | 1 | 4 15 . . .| -- | -- | -- | -- |15 . . .| -- | 1 | 2 | 9 16 . . .| -- | -- | -- | 1 |16 . . .| -- | -- | -- | 2 17 . . .| -- | -- | -- | -- |17 . . .| -- | -- | -- | 4 18 . . .| -- | -- | -- | 1 |18 . . .| -- | -- | -- | 2 19 . . .| -- | -- | -- | 1 |19 . . .| -- | -- | -- | 1 20 . . .| -- | -- | -- | -- |20 . . .| -- | -- | -- | 1 21 . . .| -- | -- | -- | -- |21 . . .| -- | -- | -- | 1 22 . . .| -- | -- | -- | -- |22 . . .| -- | -- | -- | -- --------+------+-----+-----+--------|--------|------+-----+-----+------- Total | | | | | | | | | Papers | 156 | 184 | 119 | 111 | | 156 | 187 | 118 | 111 =======================================================================

If the tests had been given in the fourth or the third grade, it would have been found that there were children, even as low as the third grade, who could do as well or better than some of the children in the eighth grade. Such comparisons of achievements among children in various subjects ought to lead at times to reorganizations of classes, to the grouping of children for special instruction, and to the rapid promotion of the more capable pupils.

In many of these measurements it will be found helpful to describe the group by naming the point above and below which half of the cases fall. This is called the median. Because of the very common use of this measure in the current literature of education, it may be worth while to discuss carefully the method of its derivation.[30]

[31]The _median point_ of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller. When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls. If the measuring instrument has been carefully derived and accurately scaled, however, it is often desirable, especially where the group being considered is reasonably large, to locate the exact point within the step on which the median falls. If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place.

The best rule for locating the median point of a distribution is to _take as the median that point on the scale which is reached by counting out one half of the measures_, the measures being taken in the order of their magnitude. If we let _n_ stand for the number of measures in the distribution, we may express the rule as follows: Count into the distribution, from either end of the scale, a distance covered by *_n/2_ measures. For example, if the distribution contains 20 measures, the median is that point on the scale which marks the end of the 10th and the beginning of the 11th measure. If there are 39 measures in the distribution, the median point is reached by counting out 19-1/2 of the measures; in other words, the median of such a distribution is at the mid-point of that fraction of the scale assigned to the 20th measure.

The _median step_ of a distribution is the step which contains within it the median point. Similarly, the _median measure_ in any distribution is the measure which contains the median point. In a distribution containing 25 measures, the 13th measure is the median measure, because 12 measures are greater and 12 are less than the 13th, while the 13th measure is itself divided into halves by the median point. Where a distribution contains an even number of measures, there is in reality no median measure but only a median point between the two halves of the distribution. Where a distribution contains an uneven number of measures, the median measure is the (_n_+1)/2 measurement, at the mid-point of which measure is the median point of the distribution.

Much inaccurate calculation has resulted from misguided attempts to secure a _median point_ with the formula just given, which is applicable only to the location of the _median measure_. It will be found much more advantageous in dealing with educational statistics to consider only the median point, and to use only the _n_/2 formula given in a previous paragraph, for practically all educational scales are or may be thought of as continuous scales rather than scales composed of discrete steps.

The greatest danger to be guarded against in considering all scales as continuous rather than discrete, is that careless thinkers may refine their calculations far beyond the accuracy which their original measurements would warrant. One should be very careful not to make such unjustifiable refinements in his statement of results as are often made by young pupils when they multiply the diameter of a circle, which has been measured only to the nearest inch, by 3.1416 in order to find the circumference. Even in the ordinary calculation of the average point of a series of measures of length, the amateur is sometimes tempted, when the number of measures in the series is not contained an even number of times in the sum of their values, to carry the quotient out to a larger number of decimal places than the original measures would justify. Final results should usually not be refined far beyond the accuracy of the original measures.

It is of utmost importance in calculating medians and other measures of a distribution to keep constantly in mind the significance of each step on the scale. If the scale consists of tasks to be done or problems to be solved, then "doing 1 task correctly" means, when considered as part of a continuous scale, anywhere from doing 1.0 up to doing 2.0 tasks. A child receives credit for "2 problems correct" whether he has just barely solved 2.0 problems or has just barely fallen short of solving 3.0 problems. If, however, the scale consists of a series of productions graduated in quality from very poor to very good, with which series other productions of the same sort are to be compared, then each sample on the scale stands at the middle of its "step" rather than at the beginning.

The second kind of scale described in the foregoing paragraph may be designated as "scales for the _quality_ of products," while the other variety may be called "scales for _magnitude_ of achievement." In the one case, the child makes the best production he can and measures its quality by comparing it with similar products of known quality on the scale. Composition, handwriting, and drawing scales are good examples of scales for quality of products. In the other case, the scales are placed in the hands of the child at the very beginning, and the magnitude of his achievement is measured by the difficulty or number of tasks accomplished successfully in a given time. Spelling, arithmetic, reading, language, geography, and history tests are examples of scales for quantity of achievement.

Scores tend to be more accurate on the scales for magnitude of achievement, because the judgment of the examiner is likely to be more accurate in deciding whether a response is correct or incorrect than it is in deciding how much quality a given product contains. This does not furnish an excuse for failing to employ the quality-of-products scales, however, for the qualities they measure are not measurable in terms of the magnitude of tasks performed. The fact appears, however, that the method of employing the quality-of-products scales is "by comparison" (of child's production with samples reproduced on the scale), while the method of employing the magnitude-of-achievement scales is "by performance" (of child on tasks of known difficulty).

In this connection it may be well to take one of the scales for quality of products and outline the steps to be followed in assigning scores, making tabulations, and finding the medians of distributions of scores.

When the Hillegas scale is employed in measuring the quality of English composition, it will be advisable to assign to each composition the score of that sample on the scale to which it is nearest in merit or quality. While some individuals may feel able to assign values intermediate to those appearing on the Hillegas scale, the majority of those persons who use this scale will not thereby obtain a more accurate result, and the assignment of such intermediate values will make it extremely difficult for any other person to make accurate use of the results. To be exactly comparable, values should be assigned in exactly the same manner.

The best result will probably be obtained by having each composition rated several times, and if possible, by a number of different judges, the paper being given each time that value on the Hillegas scale to which it seems nearest in quality. The final mark for the paper should be the median score or step (not the median point or the average point) of all the scores assigned. For example, if a paper is rated five times, once as in step number five (5.85), twice as in step number six (6.75), and twice as in step number seven (7.72), it should be given a final mark indicating that it is a number six (6.75) paper.

After each composition has been assigned a final mark indicating to what sample on the Hillegas scale it is most nearly equal in quality, proceed as follows:

Make a distribution of the final marks given to the individual papers, showing how many papers were assigned to the zero step on the scale, how many to step number one, how many to step number two, and so on for each step of the scale. We may take as an example the distribution of scores made by the pupils of the eighth grade at Butte, Montana, in May, 1914.

No. of papers 1 9 32 39 43 22 6 2
Rated at 0 1 2 3 4 5 6 7 8 9

All together there were 154 papers from the eighth grade, so that if they were arranged in order according to their merit we might begin at the poorest and count through 77 of them (n/2 = 154/2 = 77) to find the median point, which would lie between the 77th and the 78th in quality. If we begin with the 1 composition rated at 0 and count up through the 9 rated at 1 and the 32 rated at 2 in the above distribution, we shall have counted 42. In order to count out 77 cases, then, it will be necessary to count out 35 of the 39 cases rated at 3.

Now we know (if the instructions given above have been followed) that the compositions rated at 3 were so rated by virtue of the fact that the judges considered them nearer in quality to the sample valued at 3.69 than to any other sample on the scale. We should expect, then, to find that some of those rated at 3 were only slightly nearer to the sample valued at 3.69 than they were to the sample valued at 2.60, while others were only slightly nearer to 3.69 than they were to 4.74. Just how the 39 compositions rated on 3 were distributed between these two extremes we do not know, but the best single assumption to make is that they are distributed at equal intervals on step 3. Assuming, then, that the papers rated at 3 are distributed evenly over that step, we shall have covered .90 (35/39 = .897 = .90) of the entire step 3 by the time we have counted out 35 of the 39 papers falling on this step.

It now becomes necessary to examine more closely just what are the limits of step 3. It is evident from what has been said above that 3.69 is the middle step 3 and that step 3 extends downward from 3.69 halfway to 2.60, and upward from 3.69 halfway to 4.74. The table given below shows the range and the length of each step in the Hillegas Scale for English Composition.

THE HILLEGAS SCALE FOR ENGLISH COMPOSITION

======================================================
STEP No.|VALUE or SAMPLE|RANGE OF STEP |LENGTH OF STEP
--------+---------------+--------------+--------------
0. . . .| 0 | 0- .91[32] | .91
1. . . .| 1.83 | .92-2.21 | 1.30
2. . . .| 2.60 |2.22-3.14 | .93
3. . . .| 3.69 |3.15-4.21 | 1.07
4. . . .| 4.74 |4.22-5.29 | 1.08
5. . . .| 5.85 |5.30-6.30 | 1.00
6. . . .| 6.75 |6.30-7.23 | .93
7. . . .| 7.72 |7.24-8.05 | .81
8. . . .| 8.38 |8.05-8.87 | .82
9. . . .| 9.37 |8.88- |
======================================================

From the above table we find that step 3 has a length of 1.07 units. If we count out 35 of the 39 papers, or, in other words, if we pass upward into the step .90 of the total distance (1.07 units), we shall arrive at a point .96 units (.90 × 1.07 = .96) above the lower limit of step 3, which we find from the table is 3.15. Adding .96 to 3.15 gives 4.11 as the median point of this eighth grade distribution.

The median and the percentiles of any distribution of scores on the Hillegas scale may be determined in a manner similar to that illustrated above, if the scores are assigned to the individual papers according to the directions outlined above.

A similar method of calculation is employed in discovering the limits within which the middle fifty per cent of the cases fall. It often seems fairer to ask, after the upper twenty-five per cent of the children who would probably do successful work even without very adequate teaching have been eliminated, and the lower twenty-five per cent who are possibly so lacking in capacity that teaching may not be thought to affect them very largely have been left out of consideration, what is the achievement of the middle fifty per cent. To measure this achievement it is necessary to have the whole distribution and to count off twenty-five per cent, counting in from the upper end, and then twenty-five per cent, counting in from the lower end of the distribution. The points found can then be used in a statement in which the limits within which the middle fifty per cent of the cases fall. Using the same figures that are given above for scores in English composition, the lower limit is 2.64 and the limit which marks the point above which the upper twenty-five per cent of the cases are to be found is 5.08. The limits, therefore, within which the middle fifty per cent of the cases fall are from 2.64 to 5.08.

It is desirable to measure the relationship existing between the achievements (or other traits) of groups. In order to express such relationship in a single figure the coefficient or correlation is used. This measure appears frequently in the literature of education and will be briefly explained. The formula for finding the coefficient of correlation can be understood from examples of its application.

Let us suppose a group of seven individuals whose scores in terms of problems solved correctly and of words spelled correctly are as follows:[33]

======================================
INDIVIDUALS|No. OF |No. OF WORDS
MEASURED |PROBLEMS|SPELLED CORRECTLY
CORRECTLY | |
-----------+--------+-----------------
A | 1 | 2
B | 2 | 4
C | 3 | 6
D | 4 | 8
E | 5 | 10
F | 6 | 12
G | 7 | 14
======================================

From such distributions it would appear that as individuals increase in achievement in one field they increase correspondingly in the other. If one is below or above the average in achievement in one field, he is below or above and in the same degree in the other field. This sort of positive relationship (going together) is expressed by a coefficient of +1. The formula is expressed as follows:

(Sum x · y)
r = ------------------------------
(sqrt(Sum x^2))(sqrt(Sum y^2))

Here _r_ = coefficient of correlation.

_x_ = deviations from average score in arithmetic (or difference between score made and average score).

_y_ = deviations from average score in spelling.

Sum = is the sign commonly used to indicate the algebraic sum (_i.e._ the difference between the sum of the minus quantities and the plus quantities).

_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 | x·y |
--+------+---+------------+-------+---+-------------+-------------+
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 x·y = +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 x·y = -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 quantities. (A plus quantity multiplied by a plus quantity or a minus quantity multiplied by a minus quantity gives us a plus quantity as the product, while a plus quantity multiplied by a minus quantity gives us a minus quantity as the product.)

(Sum x·y) -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 | x·y=0
| AV.=4 | | | AV.=8 | | |
===================================================================

(Sum x·y) 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 |Sx·y=50
G |7 | +3 |9 |14 | +6 | 36 |
|Av. =4| |Sum x^2 =28|Av. = 8 | |Sum y^2= 112|
===============================================================

Sum x·y +50
r= -------------------------- = ---- = +.89.
sqrt(Sum x^2)sqrt(Sum y^2) 56

Other illustrations 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 classes whose scores on the Courtis addition tests were as follows:

Class 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.

Class 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