The four levels of measurements

1. Explain briefly how you would use number properties to describe the four levels of measurements.

Answer: Measurements can be classified into four different types of scales. These are:

1. Nominal
2. Ordinal
3. Interval
4. Ratio

Nominal scale:

Nominal measurement consists of assigning items to groups or categories. No quantitative information is conveyed and no ordering of the items is implied. Religious preference, race, and sex are all examples of nominal scales. Frequency distributions are usually used to analyze data measured on a nominal scale. Categorical data and numbers that are simply used as identifiers or names represent a nominal scale of measurement. Numbers on the back of a baseball jersey and social security number are examples of nominal data.

At the nominal scale, i.e., for a nominal category, one uses labels; for example, rocks can be generally categorized as igneous, sedimentary and metamorphic. For this scale some valid operations are equivalence and set membership. Nominal measures offer names or labels for certain characteristics.

The central tendency of a nominal attribute is given by its mode; neither the mean nor the median can be defined.

Ordinal scale:

An ordinal scale is a measurement scale that assigns values to objects based on their ranking with respect to one another. For example, a doctor might use a scale of 0-10 to indicate degree of improvement in some condition, from 0 (no improvement) to 10 (disappearance of the condition).

An ordinal scale of measurement represents an ordered series of relationships or rank order. Individuals competing in a contest may be fortunate to achieve first, second, or third place. First, second, and third place represent ordinal data.

In this scale type, the numbers assigned to objects or events represent the rank order (1st, 2nd, 3rd etc.) of the entities assessed. An example of ordinal measurement is the results of a horse race, which say only which horses arrived first, second, third, etc. but include no information about times.:

The central tendency of an ordinal attribute can be represented by its mode or its median, but the mean cannot be defined.

Interval scale:

Quantitative attributes are all measurable on interval scales, as any difference between the levels of an attribute can be multiplied by any real number to exceed or equal another difference. A highly familiar example of interval scale measurement is temperature with the Celsius scale. In this particular scale, the unit of measurement is 1/100 of the difference between the melting temperature and the boiling temperature of water at atmospheric pressure. The “zero point” on an interval scale is arbitrary; and negative values can be used. The formal mathematical term is an affine space (in this case an affine line). Variables measured at the interval level are called “interval variables” or sometimes “scaled variables” as they have units of measurement.

Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly. But ratios of differences can be expressed; for example, one difference can be twice another.

The central tendency of a variable measured at the interval level can be represented by its mode, its median, or its arithmetic mean. Statistical dispersion can be measured in most of the usual ways, which just involved differences or averaging, such as range, inter quartile range, and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as studentized range or coefficient of variation. More subtly, while one can define moments about the origin, only central moments are useful, since the choice of origin is arbitrary and not meaningful. One can define standardized moments, since ratios of differences are meaningful, but one cannot define coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment

Ratio scale:

The ratio scale of measurement is the most informative scale. It is an interval scale with the additional property that its zero position indicates the absence of the quantity being measured. You can think of a ratio scale as the three earlier scales rolled up in one.

The ratio scale of measurement is similar to the interval scale in that it also represents quantity and has equality of units. However, this scale also has an absolute zero (no numbers exist below the zero).

A ratio scale is a measurement scale in which a certain distance along the scale means the same thing no matter where on the scale you are, and where “0” on the scale represents the absence of the thing being measured.

Most measurement in the physical sciences and engineering is done on ratio scales. Mass, length, time, plane angle, energy and electric charge are examples of physical measures that are ratio scales. The scale type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind. Informally, the distinguishing feature of a ratio scale is the possession of a non-arbitrary zero value. For example, the Kelvin temperature scale has a non-arbitrary zero point of absolute zero, which is denoted 0K and is equal to -273.15 degrees Celsius. This zero point is non arbitrary as the particles that compose matter at this temperature have zero kinetic energy.

All statistical measures can be used for a variable measured at the ratio level, as all necessary mathematical operations are defined. The central tendency of a variable measured at the ratio level can be represented by, in addition to its mode, its median, or its arithmetic mean, also its geometric mean or harmonic mean. In addition to the measures of statistical dispersion defined for interval variables, such as range and standard deviation, for ratio variables one can also define measures that require a ratio, such as studentized range or coefficient of variation.

2. Define the terms direct measurement and indirect measurement. Describe briefly how you would make profit of indirect measurement in psychological traits.

Answer: There are 2 types of measurement techniques are developed in order to measure quality or characteristics of attributes. First one is quantitative and second is qualitative. Quantitative can be measured directly and qualitative can not be measured directly. The height and weight of a person can be measured directly with scales in feet/meter, kilogram. But qualitative variable cannot be measured with scales such as feet, meter, kilogram etc. For example, Kindness, love and intelligence of a person can not be measured directly. Indirect measurement can be used for these cases. To measure this type of cases different indirect measures like answer to questions, IQ tests can be used. Indirect measurements are mostly used in social science. Richness, happiness, good life, poverty etc can be measured with the support of different indirect indicators.

In order to measure psychological traits we use behaviors as a basis for measurement. Qualities of an individual can be measured indirectly through psychological testing by developing indicators. In standard psychological test we develop the set of standard as questionnaire or guidance fro scoring the attributes or traits. We largely use objective types of question and interpret according to the guidance of answering. Human behavior can not measure as physical measurement like height, weight. The qualitative aspects like perception, emotion, retention etc can be measured through indirect measurement, which is based on some pre-defined set of standards.

3. What will happen if you use ordinary measurement as though they were interval or ratio measurement?

Ordinary data is non parametric data and interval and ratio are parametric data. Therefore we don’t use ordinary measurement if the data are in interval or ratio measurement. They differ from each other. To ensure measurement more reliable, selection of appropriate statistical tools according to the nature of data is important. If we use interval/ratio measurement when the data are ordinal scales it may leads false decision.

4. Which method census or sampling do you prefer the most for describing the reality of Nepali classroom teaching learning? Explain in brief.

Answer: Sampling method is more applicable than the census method for describing the reality of Nepali classroom teaching learning. To study about promotion, failure and drop out rate, census method can be used. However for the reality presentation, census method can not be convenient.

Through the census method each and every unit of the population can be taken into consideration. But it will be highly time and money consuming. Sampling method will make all process faster with less cost. While taking the sample size there is more important of inclusion and representation in the sampling i.e. ethnic group, caste, religion, , geographic zone, and gender, etc. Through educational perspective different grades, private and public school/college suppose to be included. The sample size should more representatives.

5. in a group of 50 children, the 8 children who took longer than 3 hours to complete a performance test in sent-up test were marked as DNC (did not complete). In computing a measure of central tendency for this distribution of scores, what measure we should use and why?

Median can be used in computing a measure of central tendency for the distribution of score as mentioned in the question. Median is not affected by extreme values. Arithmetic mean is affected by extreme values. As Median is the positional average, we can get the correct value of central tendency.

6. Give some examples where you need geometric and harmonic mean. Give geometrical interpretation of A.M., G.M. and H.M.

Answer:

Geometric Mean (G.M):

Geometric Mean (G.M) is widely used in averaging ratios and percentages and is computing average rates of increase or decrease. It is also advantageously used in the construction of index numbers. G.M. gives equal weights to equal ratios of change. It is also used to compute the average rate of growth or reduction of population or average increase or decrease of production, profit, sales etc. When we require to give more weight to smaller items and smaller weight to larger (e.g. Social and economic problems) G.M can be used.

Harmonic Mean (H.M.):

Harmonic Mean (H.M.) is used in computing the averages relating to the rates and ratios such as velocity speed etc., where time factor is the variable. It also can be used for making Human Development Indicator (HDI).

Geometrical interpretation of A.M., G.M., and H.M.

Let AD = a, DB = b

Then represents the radius of the semi – circle.

Hence radius OP = , which gives the value of A.M.

Similarly radius OQ = , Now OD = b =

Now DQ2 = OQ2 – OD2 = { }2 – { }2 = ab

Hence, DQ = , which represents G.M.

Now, in the right angled triangle ODM, DM2 = OD2 – OM2

And in right angled triangle DMQ, DM2 = DQ2 – MQ2

Hence, OD2 – OM2 = DQ2 – MQ2

Here, OQ = . Let OM = x, then MQ = – x

{}2 – x2 = ab – { – x}2

For solving, x =

Hence, MQ = – = , which represents H.M.

From above it is clear that OP = A.M.,

DQ = G.M.

MQ = H.M.

From the figure, it is clear that OP > DQ > MQ.

Hence, we can say that A.M. > G.M. > H.M.H

7. Give geometrical meaning of the formula used for Median and Mode for grouped data.

Answer.: Geometrical meaning of the formula used for Median:

Let consider the following continuous frequency distribution, (x1 < x2 < ……xn+1).

Class interval: x1 – x2 , x2 – x3, …………………. xk – xk+1, ………. xn – xn+1

Frequency: f1 f2 …………………… fk …………… fn

The cumulative frequency distribution is given by:

Class interval: x1 – x2 , x2 – x3, …………………. xk – xk+1, ………. xn – xn+1

frequency : F1 F2 …………………… Fk …………… Fn

Where, Fi = f1 + f2 + ………..+ fi-1. The class xk – xk+1 is the median class if and only if

Fk-1 < N/2 < Fk.

Now, if we assume that the variate values are uniformly distributed over the median – class which implies that the ogive is a straight line in the median – class, then we get from the fig.1,

tan =

i.e.

or

or,

=

Where is the frequency and h the magnitude of the median class.

Hence, BS =

Hence, Median = OT = OP + PT = OP + BS = l +

This is the required formula.

Geometrical meaning of the formula used for Median:

Let us consider the continuous frequency distribution:

Class interval : x1 – x2 , x2 – x3, …………………. xk – xk+1, ………. xn – xn+1

frequency : f1 f2 …………………… fk …………… fn

If fk is the maximum of all the frequencies, then the modal class is (xk – xk-1).

Let us further consider a portion of the histogram, namely, the rectangle erected on the modal class and the two adjacent classes. The modal is the value of x for which the frequency curve has a maxima. Let the modal point be Q (fig. 2)

From the figure, we have tanß =

and tana =

or,

or, , where h is the magnitude of the model class. Thus solving for LM, we get

LM =

Hence, Mode = OQ = OP + PQ = OP + LM

= l +

8. Squaring deviations and then taking squares seems to be useless. Why do we use square?

Answer: Squaring deviation and then taking squares seems to be useless however actually it has certain meaning like the squaring of the deviations (x-x) removes the drawbacks of ignoring signs of the deviations in computation of mean deviation. Taking the sign into consideration we obtain positive values always when squared. But squaring gives aunit that isthe square of theunit the quantity is measured in. This step provides it suitable for further mathematical treatment.

9. Study the following summary statistics of the scores of two graders VI and VII. Now give your answer to the following questions and give figures to support your answers.

a. Which class had the larger number of pupils?

Answer: Grade VI had larger number of pupils.

b. Which class on the average had the higher scores?

Answer: Grade VII on the average had the higher scores.

c. In which class were the scores more scattered? (Given four different statistics to show the difference in scatter.)

Answer: For Grade VI, the scores are more scattered. The four different measures to show the difference in scatter ness are as follows:

1. Interquartile range
2. Coefficient of S.D.
3. Coefficient of M.D. from mean
4. Coefficient of variation

9. Are the distributions of scores about the mean symmetrical? What is your evidence? If not, which class has high scores not balanced by similar low scores?

The distribution of scores about the mean in both classes are not symmetrical as we can find Mean = Median = Mode is not satisfied for both the grades.

In grade VI, since Mean < Median <Mode, it is negatively skewed. That is there is greater variation towards the lower values of the variable.

In grade VII, since Mean > Median > Mode, it is positively skewed. That is there is greater variation towards the higher values of the variables.

10. Take one distributed data grouped into different frequencies and calculate different measure of central tendencies (Arithmetic mean, Median, and mode) and measures of dispersion (Q.D., MD, and SD). Give your judgments about your data concerning to symmetry.

Answer: Suppose, the weights of 50 students of a class are classified below.

For Mean;

Mean = A +

= 65- = 64.87

Hence, Mean =64.87

For Median;

Hence, Median lies in the class 60-70

Median = = 60+=66.2

Hence, Median=66.2

For Mode;

Since maximum frequency occurs at two classes, so the given distribution is a bimodal distribution.

So, Mode =3 median-2 mean

=3*66.87-2*64.87

=198.6-129.74=68.86

Hence, Mode=68.86

For Quartile Deviation;

Position of Q1=

Hence First Quartile (Q1) lies in the interval 50-60

Now, Q1 =

50+

Hence, First Quartile (Q1)= 57.28

Position of Q3=

Hence Third Quartile (Q3) lies in the interval 70-80

Again, Q3 =

=70+

=73.62

Now, QD= ==8.17

Hence, Quartile Deviation (QD) = 8.17

For Mean Deviations;

Mean deviation from mean

Calculation of Standard Deviations

Now, N = 75, Σfd’ = -1 Σfd’2 = 89

=

=

=1.08*10

=10.81

To identify Symmetry

Here, Mean = 64.87

Median = 66.2

Mode = 68.86

Hence, the curve is not symmetrical.

Calculation of Skewness

Sk = is negative skewed.