College of Education

School of Continuing and Distance Education 2014/2015 – 2016/2017

PSYC 331 STATISTICS FOR

PSYCHOLOGISTS

Session 1 – BASIC CONCEPTS IN

STATISTICS

Lecturer: Dr. Paul Narh Doku, Dept of Psychology, UG Contact Information: pndoku@ug.edu.gh

godsonug.wordpress.com/blog

Session Overview

• This session introduces you to an review of some basic statistical

concepts that will enable you to understand the

principles in inferential statistics.

•By the end of this session learners should be able to

concepts and

clearly distinguish between descriptive and inferential statistics,

have a good grasp of some summation notation,

have a comprehensive understanding of the different scales of measurement used in social and educational research and their relation to statistical test and finally

clearly understand what is meant by the mean, standard

deviation and variance of a set of numbers or observations.

Dr. P. N. Doku, Slide 2

Session Outline

The key topics to be covered in the session are as follows: •Topic one - Descriptive and inferential statistics

•Topic Two- Summation notation

•Topic Three - Measurement and scales of measurement in the Social Sciences and Education

•Topic Four - Scales of measurement and their applications to statistical tests •Topic Five - The mean, standard deviation and variance of data measured on an interval or ratio scale

•Topic Six - Calculating the standard deviation and variance from raw scores

Slide 3

Reading List

• Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 3 – 22

• MiĐhael Baƌƌoǁ, StatistiĐs foƌ EĐoŶoŵiĐs, AĐĐouŶtiŶg aŶd BusiŶess Studies, 4th Edition, Pearson

• R.D. Mason , D.A. Lind, and W.G. Marchal, StatistiĐal TeĐhŶiƋues iŶ BusiŶess aŶd EĐoŶoŵiĐs, ϭϬth Edition, McGraw- Hill

Slide 4

5

TOPIC ONE: DESCRIPTIVE

AND INFERENTIAL

STSTISTICS WHAT IS STATISTIC?

•Statistic is a very broad

subject, with applications

in a vast number of different fields. Statistics

has TWO meanings; namely,

a discipline/subject and

computation based on sample data (a sample calculation)

Descriptive and inferential statistics

• There are two major types of statistics (viewed as a sample calculation).

Descriptive Statistics

Inferential Statistics.

• The branch of statistic devoted to the

summarization and description of data

is called DESCRIPTIVE STATISTICS

(data spread and central tendency)

• and the branch of statistics concerned

with using sample data to make an

inference about a population of data

is called INFERENTIAL STATISTICS. Slide 6

7

Key Terminologies: The concept of Population and Sample

• Population and sample are two basic concepts of statistics.

• A POPULATION may be defined as the totality/

collection of all objects, animals (including human beings), things, events, or even happenings.

• Example, all students following academic programmes at the University of Ghana make up the population of students at the University of Ghana.

• But often only a subset(small portion) of individuals of that population may be observed; such a subset of individuals constitutes a SAMPLE.

• Example, students offering Psychology in University of Ghana will constitute a sample for the above population.

The concept of sample

• A sample is a part of a population. In the example I gave above on the population of students at the

University of Ghana, the total number of students

studying under distance learning forms part of the

total number of students following academic

programmes at the University of Ghana and hence

will be a sample based on the population of students at the University of Ghana.

Descriptive statistics Descriptive statistics are generally

procedures for summarizing and describing

quantitative information or data. In

descriptive statistics

Descriptive statistics includes the

construction of graphs, charts, and tables, and the calculation of various descriptive

measures such as averages, measures of variation, and percentiles.

In descriptive statistics, We cannot tell exactly what is happening in the population

from the description BUT can make guesses

of what may be happening in the population

because only part of the population is

studied when we take a sample from a

population.

Inferential Statistics

Inferential statistics is solely concerned about techniques of drawing inferences (conclusions) from a sample to a population.

As an example, suppose that you were interested in determining the average annual income of all residents in Accra, Ghana. Let us assume that there are ten million (10,000,000) households in Accra. The 10,000,000 households represent the population of households in Accra. It will obviously not only be expensive but also time-consuming to go to every household and determine the income levels of the occupants. Instead, you might obtain a list of all households in Accra and from this list randomly select about 500 households, visit the selected households and obtain the income levels of the occupants of each household. The 500 selected households represent a sample from the population of all households in Accra. From this data, you can calculate the mean (average) annual income and standard deviation based on the sample of 500 households.

Inferential Statistics cont. Let us assume that you find that the average annual income

based on your sample is (GH¢300.00) with a standard deviation of (GH¢30.00). From this data, you can then say that given a certain margin of error (which is very small if the sample was carefully selected from the population), then the population mean and standard deviation are also approximately equal to GH¢300.00 and GH¢30.00 respectively.

Thus, you have made an inference or drawn a conclusion from a sample to a population. Note that this inference (estimate of population mean and

standard deviation from the sample data) can be either right or wrong depending on how your sample was selected (only rich people or poor people ).

It will be a biased sample and the sample mean and standard deviation will not be correct estimates of the population.

Sample statistics • All computations made based on sample information is called a

sample statistic.

• In the example above, the average annual income of GH¢300.00

and standard deviation of GH¢30.00 are estimates based on a

sample.

• Usually denoted by ENGLISH ALPHABETS (e.g. for sample mean; s

for sample standard deviation; S2 for sample variance).

• You must remember then that anytime you come across English

alphabets, it means that in most (but not in all) situations you are

dealing with a sample.

s 2

Population parameter • Estimations based on the entirety of the population

becomes the population parameter.

• In the previous example, if you had studied the whole

population of 10,000,000 households, then the average

annual income and standard deviation that you calculate

would be estimates based on a population.

• Usually symbolized by small GREEK ALPHABETS (e.g., for population mean; for population standard deviation; 2

for population variance).

• In effect, inferential statistics that this course is also about is

concerned about techniques used in drawing inferences

from sample statistics to their corresponding population

parameters.

Dr. Richard Boateng, UGBS Slide 14

Dr. Richard Boateng, UGBS Slide 15

SUMMATION NOTATION Some notations that are frequently used when you are adding up scores or data collected for statistical analysis. These notations include •Sum of a set of numbers or observations •Sum of squares of a set of numbers or observations •Square of the sum of a set of numbers or observations •Sum of the product of two sets of numbers or observations •Sum of squares of the product of two sets of numbers or observations •Square of the sum of the product of two sets of numbers or observations

Σ ………. Sigŵa …………ŵeaŶiŶg suŵŵatioŶ

Sum of a set of numbers or observations

• The capital Greek letter Σ , ;pƌoŶouŶĐed sigŵaͿ, staŶds foƌ suŵ of iŶ statistiĐs. Suppose you ǁaŶt to fiŶd the suŵ

of the following numbers: 2, 4, 6, 8, and 12. If you let represent any number, then you

can refer to the five numbers listed above as and , where the subscripts 1, 2, 3, 4, and 5 stand for the ordinal positions of the numbers (the

order in which the numbers are written).

Sum of a set of numbers or observations

• In the above example, and . In

general, if there are n numbers in a set of measurements, then

the sum of the n numbers can be symbolically written as

• This notation simply means that you add up all the X values, starting from the first value of X, to the last value of X, . In

our example above, i ranges from 1 to 5 and therefore can

simply be written as

.

Sum of a set of numbers or observations

• When there is no confusion, the subscript i and the superscript n in the

notation may be dropped and therefore the sum of the above

set of numbers may simply be written as

Sum of squares of a set of n numbers or observations

• Sum of squares means that each X value (number) as in the previous example must first be squared and the resulting values added up.

• Symbolically, the sum of squares of n numbers is written as

or simply as .

• In the previous example in the preceding slide involving the five numbers 2, 4, 6, 8, and 12,

= 4 + 16 + 36 + 64 + 144 = 264

Square of the sum of a set of n numbers or observations

• Square of the sum means that all the X values must first be added up and the resulting sum squared.

• Symbolically, the square of the sum of n numbers is

written as or simply as .

• In the illustrations so far involving the five

numbers, 2, 4, 6, 8, and 12, =

=

= 1,024

Sum of the product of two sets of numbers or observations (X and Y)

• In social science research, you may sometimes be interested

in finding out the relationship between two variables. (A

variable is any quantity that changes among individuals or over time, for example, grades obtained by students in an

examination).

• OŶe ǀaƌiaďle Đould ďe PeƌfoƌŵaŶĐe iŶ MatheŵatiĐs aŶd the seĐoŶd ǀaƌiaďle, PeƌfoƌŵaŶĐe iŶ StatistiĐs. You ŵay haǀe a prediction that students who are good in Mathematics are

also good in Statistics and students who are poor in

Mathematics are also poor in Statistics.

Sum of the product of two sets of numbers or observations (X and Y) • Let us ƌepƌeseŶt oŶe ǀaƌiaďle, say PeƌfoƌŵaŶĐe iŶ

MatheŵatiĐs ďy the syŵďol X aŶd let us ƌepƌeseŶt the otheƌ ǀaƌiaďle, PeƌfoƌŵaŶĐe iŶ StatistiĐs ďy the syŵďol Y. IŶ this example, let us assume that examination performance in

both Mathematics and Statistics is marked over 15.

• For simplicity, let us assume that five students took the

examination in both Mathematics and Statistics and that the

X values (performance in Mathematics) were 2, 4, 6, 8, and

12, and that the Y values (performance in Statistics) for the

same students were also 3, 4, 5, 7, and 10 respectively.

• Please note that for each X value there must be a

corresponding Y value since each student was assessed on

both the X and Y variables.

Sum of the product of two sets of numbers or observations (X and Y)

The above data can be arranged in a table as:

STUDENT NO. Score in Maths(X) Score in Stats(Y) XY

1. 2 3 6

2. 4 4 16

3. 6 5 30

4. 8 7 56

5. 12 10 120

Totals ΣXY=228

Sum of the product of two sets of numbers or observations (X and Y) • Alternatively the sum of the product of the X and Y

values could be symbolically written as ΣXY and

thus solve accordingly as:

= 6 + 24 + 30 + 56 + 120

= 228

Slide 25

Sum of squares of the product of two sets of numbers or observations (X and Y)

The sum of squares of the product of two sets of numbers or observations, X and Y, is symbolically written as .

•That is, we multiply the corresponding X and Y values, square

each obtained product and add up the resulting products. In

the example given so far:

=

= 36 + 256 + 900 + 3,136 + 14,400

= 18,728

Square of the sum of the product of two

sets of observations (X and Y)

The square of the sum of the product of two sets of numbers

or observations (X and Y) is symbolically written as .

•That is, we first find the sum of the product of the two sets of numbers as in the previous example above and square this

sum. Using

the same figures in the previous examples

=

=

= 2282

= 51,984

MEASUREMENT AND SCALES OF MEASUREMENT IN THE

SOCIAL SCIENCES AND EDUCATION

• A measurement scale can possibly possess three

properties: the properties of magnitude, equal intervals and an absolute zero point.

• Measurement in the Social Sciences and

Education may range from physical and

behavioural measures (e.g., height, weight, distance, etc., for physical measurements; and

anxiety, depression, self- esteem, etc., for behavioural measurements); through rankings

of physical and behavioural measures (e.g. 1st , 2nd , 3rd , etc.); to mere classification of objects, events, or happenings (e.g., male/female, present/absent, Yes/No, etc.).

• The type of measurement adopted represents the

scale of measurement.

Properties of Scales of Measurement

Scales of measurement in social and behavioural research have

three(3) basic properties:

•The property of Magnitude

•The property of Equal Intervals

•The property of Absolute Zero point

• Magnitude--pƌopeƌty of ŵoƌeŶess. Higher score refers to more of something.

• Equal intervals--is the difference

between any two adjacent numbers

referring to the same amount of difference on the attribute?

• Absolute zero--does the scale have

a zero point that refers to having

none of that attribute?

Property of Magnitude

• When a scale of measurement has magnitude, it means

that an instance of whatever is being measured on the

scale can be judged greater than ( > ), less than ( < ), or equal to ( = ) another instance of what is being

measured.

• For example, 5 kilometres (5km) is greater than ( > ) a

distance of 3 km;

• If the distance between two towns, A and B , is 10

km and the distance between two towns, C and D , is also 10km, then the distance between towns A and B is equal to ( = ) the distance between towns C and D.

• Items could be ranked, 1st, 2nd, 3rd etc. Slide 30

Property of Equal Interval

• When a measurement scale has equal intervals, it means

that a unit of measurement between any two points on

the scale is the same regardless of where the two points

fall on the scale. For example, in the measurement of distance using the metre rule, the distance between 10cm

and 11 cm is the same as the distance between 75cm and

76cm on the scale.

• Please note that any measurement scale that possesses

the property of equal intervals also possesses the property

of magnitude. In the measurement of distance, (11cm

> 10cm),

Slide 31

Property of Absolute Zero Point

• When we say that a measurement scale has an absolute

zero point, it means that (in theory), at one end of the

scale, nothing at all of whatever is being measured exists. For example, we may think of a distance of 0mm (no

distance) or a weight of 0mg (a weightless object) if our units of measurement for distance and weight are

millimetres and milligrams respectively.

• We may design a scale to measure something where we

may decide to place zero (0) at one end of the scale. However, this does not necessarily mean that at that end

of the scale, nothing at all of what we are measuring

exists.

Slide 32

Property of Absolute Zero Point cont:

• For example, one thermometer that we use to measure

temperature starts from 0oC to 100oC on the Celsius scale. In more

technical language, this thermometer is calibrated from 0oC to

100oC. Does it mean that when the thermometer reading is 0oC

today, it means that there is no temperature? Absolutely no! The

point 0oC is merely chosen for convenience as the starting point of the scale to represent the melting point of ice.

• An important characteristic of a measurement scale with an

absolute zero point is that we can make ratio statements on

measurements with the scale. For example, if a

boy weighs 30kg and a man weighs 60kg, we can say that the man

weighs twice as the ďoy; oƌ the ďoy’s ǁeight is half the ŵSlidae 3Ŷ3 ’s ǁeight

Types of measurement scales

Nominal Scale

• Assigns a value to an object for identification or classification purposes.

• Most elementary level of measurement.

• Nominal is hardly measurement. It refers to quality

more than quantity.

• A nominal level of measurement is simply a matter of distinguishing by name, e.g., 1 = male, 2 = female. Even though we are using the numbers 1 and 2, they

do not denote quantity.

Nominal Scale cont:

They are categories or classifications, examples:

•MEAL PREFERENCE: Breakfast, Lunch, Dinner

•RELIGIOUS PREFERENCE: 1 = Buddhist, 2 =

Muslim, 3 = Christian, 4 = Jewish, 5 = Other

•POLITICAL ORIENTATION: Republican, Democratic, Libertarian, Green

Ordinal Scale

• Ordinal refers to order in measurement.

• Ranking scales allowing things to be arranged based on how

much of some concept they possible.

• Have nominal properties

• An ordinal scale indicates direction, in addition to providing

nominal information. Low/Medium/High; or Faster/Slower are

examples of ordinal levels of measurement. Ranking an

experience as a "nine" on a scale of 1 to 10 tells us that it was

higher than an experience ranked as a "six." Many psychological scales or inventories are at the ordinal level of measurement.

Ordinal Scale cont:

• Examples:

• RANK: 1st place, 2nd place, ... last place

• LEVEL OF AGREEMENT: No, Maybe, Yes

• POLITICAL ORIENTATION: Left, Center, Right

Interval Scale

• Capture information about differences in quantities of a concept.

• Have both nominal and ordinal properties.

• Interval scales provide information about order, and

also possess equal intervals.

• From the previous example, if we knew that the

distance between 1 and 2 was the same as that between 7 and 8 on our 10-point rating scale, then

we would have an interval scale.

Interval Scale cont:

Examples:

•TIME OF DAY on a 12-hour clock

•POLITICAL ORIENTATION: Score on standardized scale of political orientation

•OTHER scales constructed so as to possess equal intervals

•Interval time of day - equal intervals; analog (12-hr.) clock, difference between 1 and 2 pm is same as difference between 11 and 12 am

• Highest form of measurement.

• Have all the properties of interval scales with the additional attribute of representing absolute quantities.

• Absolute zero - In addition to possessing the qualities

of nominal, ordinal, and interval scales, a ratio scale has

an absolute zero (a point where none of the quality

being measured exists).

• Using a ratio scale permits comparisons such as being

twice as high, or one-half as much.

Ratio Scale

Examples:

• RULER: inches or centimeters

• YEARS of work experience

• INCOME: money earned last year

• NUMBER of children

• GPA: grade point average

Ratio Scale cont:

Scales of Measurement and Behavioural Measurements

• As you can see from the examples given, scales with absolute zero points are

mainly to be found in the Physical Sciences. Most of the scales of measurement used in the Social Sciences and Education possess only the

property of magnitude and in some cases, equal intervals .

• Take the measurement of a human trait or characteristic like aggression as

an example. In the measurement of aggression, it does not make sense to

think of a scale where nothing at all of the aggressive trait or characteristic

exists at one end of the scale.

• This is because every human being has some level of aggression within

him/her. It is a matter of degree to which individuals show this aggression. Thus, any scale designed to measure aggression, depression, anxiety, attitude toward some object, etc., will have no absolute zero point. At best, such a scale may have equal intervals, which in most cases are assumed to

be the case when in fact it may not really be so.

SCALES OF MEASUREMENT AND

THEIR RELATIONSHIPS TO STATISTICAL

TESTS • The choice of a statistical test to

analyze data will depend, to some

extent, on the measurement scale

used to collect the data. Thee are

2 broad groups of tests to analyze

data.

• Parametric statistical tests that

make assumptions about the shape

of the distribution of data in the

population are employed when the

level of measurement is either ratio or interval.

• Nonparametric or distribution-free

statistical tests are

statistical tests that do

not make any

assumptions about the

shape of the distribution

of data in the population. They are

used when the level of measurement is either ordinal or nominal.

Parametric statistical tests

Examples

•The z test

•t tests – one sample

t test, independent t test, dependent or correlated t test

•The ANOVA or F

test, and

•The Pearson

product- moment correlation

coefficient (abbreviated as

Pearson r).

Assumptions

•Ratio/Interval Data

collected on the

outcome/Dependent variable

•Normality: Data have a

normal distribution (or at least is symmetric)

•Homogeneity of variances: Data from

multiple groups have

the same variance.

•Linearity: Data have a

linear relationship.

•Independence: Data

are independent Slide 45

Non-Parametric statistical test

Examples

•Spearman

rank-order correlation

coefficient (abbreviated as

Spearman )

•Chi square

test.

•Mann-Whitney U test

• Wilcoxon matched-pairs signed- ranks test,

• Spearman Rho

• Kruskal-Wallis H test.

Slide 46

Types of Statistical tests that are used

with various scales of measurement

measurement Scale of Types of Test to use Examples of Specific Tests to use

Measurement

Ratio

Interval

Parametric

tests

DIFFERENCES:

Z test,

one sample t test,

independent t test,

dependent or correlated t test,

One way ANOVA (F) test,

RELATIONSHIP:

Pearson r

DIFFERENCES:

Ordinal

Nominal

Nonparametric

Tests

Wilcoxon matched-pairs signed- ranks test,

Kruskal-Wallis H test.

Mann-Whitney U test,

RELATIONSHIPS

Spearman

Chi Square test Slide 47

THE MEAN, STANDARD DEVIATION AND VARIANCE OF DATA MEASURED ON AN INTERVAL OR RATIO SCALE: THE MEAN

• In inferential statistics, the

mean refers to the

arithmetic mean. It is

perhaps the most widely- used measure of central tendency

• The arithmetic mean of a

set of numbers is simply

the sum of the numbers

divided by the total number of observations

Mean:

• For example, to find the mean of the following

numbers: 5, 6, 8, 10, 15, and 16, you simply find

the sum of the numbers and divide this sum by the

total number of observations, which is equal to six

(6) in this case. Thus, the mean of the above set of numbers =

=10.00.

Slide 49

Mean:

• In general, if we let Xi or simply X stand for any

number in a set of numbers, then the sum of the

numbers in the set = ΣX.

• If we let N or Ŷ stand for the total number of observations, then the mean can generally be written as

• N is normally used to stand for population size

while Ŷ is used to stand for sample size Slide 50

Mean

• The mean based

on POPULATON

data is symbolized

by

• Population mean:

• the mean based on a

sample symbolized

by

• Sample mean:

Slide 51

Measurement of variability of a set of numbers or observations

• Variability refers to the extent to which a set of numbers or observations vary about or deviate from a

measure of central tendency (the mean).

• Measures of variability are also sometimes called measures of dispersion.

• Examples include the standard deviation, the

variance, and the range

Measurement of variability of a set of numbers or observations

• As an illustration, consider the following two sets of numbers: (i) 10, 11, 11, 12, 13, 12, 10, 13; and (ii) 10, 14, 19, 23, 30, 29, 45, 50.

• If you carefully inspect the two sets of data, you will notice that the numbers in set number (i) are close

to each other in values than the numbers in set number (ii).

• This means that the numbers in set (ii) show greater variability than the numbers in set (i).

Measurement of variability of a set of numbers or observations

• Now, consider a situation where the numbers 5,6,8,10,15,16 stand for the

scores (out of 20) obtained by six students in an examination in Mathematics

with a mean score of 10. If the six students were of equal ability in all respects, then it is reasonable to expect that each student would have obtained the score

of 10, the mean score

• But it is totally unrealistic to expect that all the six students would be of equal ability due to individual differences on factors like intelligence, interest, motivation, practice, etc. For this reason, we expect that some students will obtain higher scores than others

• The reason for calculating variability is to determine the extent to which the

scores of these six students in our example, vary about or deviate from the

mean score of 10.

A practical demonstration of the

concept of variability • To find this variability or deviation, you may think that you can do this by

siŵply suďtƌaĐtiŶg eaĐh studeŶt’s sĐoƌe fƌoŵ the ŵeaŶ sĐoƌe of ϭϬ as folloǁs: (5-10); (6-10); (8-10); (10-10); (15-10); and (16-10). You may then

add up the values to obtain a measure of variability, i.e., variability = (-5) + (-4) + (-2) + (0)

+ (5) + (6) = (-11) + (11) = 0.

• You see what has happened? The figure 0 (zero) that you obtained means

that there is no variability.

• In other words, the students are all obtaining the same score! However, by

inspecting the data, we notice differences in individual scores, hence there is

variability.

• The reason why we are obtaining zero here is that in Mathematics, for any

set of numbers, the sum of the difference between the numbers and their mean is always equal to zero. [You may try proving this with another set of numbers]

The concept of variance

• A way out of the difficulty we are faced with in calculating variability using the simple logic in preceding slide will be for us to use the squares of the numbers, instead of the raw numbers, In other words, the squares of the difference between the set of scores (numbers) and their mean are used to calculate variability. Using this method, the variability of the scores recorded for the six students in our example will be:

• (5-10)2 + (6-10)2 + (8-10)2 + (10-10)2 + (15-10)2 + (16-10)2

• = (-5) 2 + (-4) 2 + (-2) 2 + (0) 2 + (5) 2 + (6) 2

• = 25 + 16 + 4 + 0 + 25 + 36

• = 106.

• The computed value of 106 is the total variability of the 6 scores about their mean

• Variance is the average variability which is 106/6 » 17.67 (corrected to 2

decimal places)

Formula for variance

• Population variance

symbolized by • Sample variance

symbolized by

Slide 57

The concept of standard deviation • Note that because deviation of each score from the mean was squared to

estimate the variance in the previous example, variance is in square units whereas

the original measurements are not in square units.

• The standard deviation is simply the square root of the variance and is preferred over the variance because:

1. Taking the square root of the variance (i.e., finding the standard deviation) brings

the measure of variability or dispersion to the same unit of measurement as the

original measurement of the ten individuals.

• For example supposing the measurement were in (cm), variance will square them

and hence (cm2); finding the square root(standard deviation brings the unit back

to cm(original unit.

2. The value of the standard deviation is normally less then the mean and being

measured in the same units as the mean, it becomes possible to roughly estimate

the spread of scores about the mean

Formula for standard deviation

• Population parameter • Sample statistic

Slide 59

CALCULATING THE STANDARD DEVIATION AND

VARIANCE FORM A RAW DATA

• It is relatively easier to calculate the

standard deviation and variance from raw

scores than calculating these values using

the deviations from the mean method just illustrated, but that the different methods

in calculating will give you the same

values.

The standard deviation and variance based on a population

•Fortunately, there is a relatively easier way of calculating the standard deviation and variance

based on the raw scores without going through

the tedious exercise involved in the calculations

using the previous formulae (deviation from the

mean).

•This easier formula which may be used for any

population and sample sizes is based on sum of squares and squares of the sum of the raw

scores.

The standard deviation and variance based on a population

Population standard deviation

To get the variance, square the standard deviation.

• Please note again that in the above

formula, for sum of squares of the raw data

while the square of the sum of the raw

scores.

stand stands for

The standard deviation and variance based on a sample

The standard deviation and variance based on a sample

is also calculated from the raw data using sum of squares

and square of the sum of the data using the following

formula:

IŶ the aďoǀe foƌŵula, s as usual stands for the standard

deviation and n stands for the sample size. Note that ǁe ĐalĐulate the saŵple statistiĐ, s, to estiŵate the corresponding population parameter, .

References

• Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 3 – 22

• MiĐhael Baƌƌoǁ, StatistiĐs foƌ EĐoŶoŵiĐs, AĐĐouŶtiŶg aŶd

BusiŶess Studies, 4th Edition, Pearson

• R.D. MasoŶ , D.A. LiŶd, aŶd W.G. MaƌĐhal, StatistiĐal TeĐhŶiƋues iŶ BusiŶess aŶd EĐoŶoŵiĐs, ϭϬth Edition, McGraw-Hill

Slide 64