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statistics is easy to learn
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Dr. Daxaben N. MehtaPrincipalSmt. S.C.U.Shah Home Science and C.U.Shah Arts & Commerce Mahila CollegeWadhwancity – Dist: Surendranagar
STATISTICS IN RESEARCH
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To understand God's thoughts we must study statistics, for these are the measure of His purpose.
— Florence Nightingale
Statistical thinking will one day be as necessary a qualification for efficient citizenship as the ability to read and write.
— H.G. Wells
Why Statistics ?
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What in Statistics
Role of Statistic
Types of Statistics
Terminology
Collecting Data
Representing Data
HypothesesTesting
Analyzing Data
Asking the Research Question
THEORY
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Role of Statistics in research
• ValidityWill this study help answer the research question?
• AnalysisWhat analysis, & how should this be interpreted and reported?
• EfficiencyIs the experiment the correct size,making best use of resources?
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A set of methods, procedures and rules for organizing, summarizing, and interpreting information.
There is a distinction between statistics and parameters
Here, it would be better to speak of statistical methods.
Statistics
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Parameters and StatisticsParameter: the value of a variable in a population.Statistic: the value of a variable in a sample.Statistics are often used to estimate or draw inferences about parameters.
Parameters and Statistics
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Statistical inference is the process of estimating population parameters from sample statistics.
Statistical inference
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10
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Hei
gh
t in
inch
es
Males Females
Are males taller than females?
Statistical inference may be used to ascertain whether differences exist
between groups...
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Variable: any characteristic that can vary across individuals, groups, or objects. For example:WeightOccupationGrade-point averageLevel of test anxiety
Variable
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Variables
1. The dependent variable is always the property you are trying to explain; it is always the object of the research.
2. The independent variable usually occurs earlier in time than the dependent variables.
3. The independent variable is often seen as influencing, directly or indirectly, the dependent variable.
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Values: the numerical value of a particular realization of a variable.For instance if the variable is weight Than a child weighs15kg then the value of the variable for child is 15.
Values
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Sampling error is the difference between a sample statistic and its corresponding population parameter.The values of sample statistics vary from sample to sample, even when all samples are drawn from the same population.
Sampling Error
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Organized arrangements of sets of data by order of magnitude or Sequential listings of data points from lowest to highest.Frequency distributions.- A sequential listing of data points combined with the number of times (or frequency with which) each point occurs.
Distributions
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Statistical procedures are the tools of research.There are several types (or methods) of research studies and the type of statistical procedure used will often vary from one type of research to another.
Statistical procedures
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The correlational method of research.Examines relationships among two or more variables. The experimental method is used when the researchers wants to establish a cause and effect relationship A quasi-experiment is similar to a (true) experiment except that here the independent variable is not manipulated by the researcher
Statistical procedures
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Another tool of [quantitative] research.Definition: A rule for the assignment of numbers to attributes or characteristics of individuals, or things
Types of measurement has implications for the type of statistical procedure employed.Some statistical procedures assume a certain level of measurement.
Measurement Scale
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Three can be distinguished: nominal, ordinal, and scale (includes interval and ratio).Nominal Coarse level of measurement used for identification purposes.Substitutes numbers for other categorical labels. No order of magnitude is implied.Examples: sex (male or female), student classification Do you have a loss of appetite?
Types of Measurement
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Ordinal.Objects measured on an ordinal scale differ from each other in terms of magnitude, but the units of magnitude are not equal. The objects can be ordered in terms of their magnitude (more or less of an attribute. Examples: socioeconomic status, level of education attained (elementary school, high school, college degree, graduate degree)
Types of Measurement
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Types of Measurement
Scale includes both interval and ratio level scales. Scale measurements yield equal intervals between adjacent scale points.The difference between score of 435 and 445 is the same as the difference between a score of 520 and 530. IQ scoresMost scores obtained form achievement tests, aptitude tests, etc. are treated as scaled data
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Interval level:What is your age in years? Ordinal level:What is your age group?
18 years or younger 19-44 years 45 years or older
Same Variable, Different Levels of Measurement
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Choosing the Appropriate Statistic
Some factors to consider:Research designNumber of groupsNumber of variablesLevel of measurement
(nominal, ordinal, interval/ratio)
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Statistical Methods
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Descriptive statistics characterize the attributes of a set of measurements. Used to summarize data, to explore patterns of variation, and describe changes over time.
Types of Statistics
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Descriptive Statistics
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Descriptive Statistics
• Tabular and Graphical Methods• Qualitative Data
• Quantitative Data
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Tabular and Graphical Procedures
DataDataQualitative DataQualitative DataQualitative DataQualitative Data
Quantitative DataQuantitative Data
TabularTabularMethodsMethods
TabularTabularMethodsMethods
GraphicalGraphical MethodsMethods
GraphicalGraphical MethodsMethods
•FrequencyFrequency DistributionDistribution
•Bar GraphBar Graph•Pie ChartPie Chart
•FrequencyFrequency DistributionDistribution•Cum. Freq. Dist.Cum. Freq. Dist.
•HistogramHistogram•OgiveOgive
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Line Graph
• The line graphs are usually drawn to represent the time series data Example: temperature, rainfall, population growth, birth rates and the death rates.
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Line Graph
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Polygraph
• Polygraph is a line graph in which two or more than two variables are shown on a same diagram by different lines. It helps in comparing the data. Examples which can be shown as polygraph are:The growth rate of different crops like rice, wheat,
pulses in one diagram.The birth rates, death rates and life expectancy in
one diagram.Sex ratio in different states or countries in one
diagram.
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Polygraph
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Qualitative Data
• Frequency Distribution
• Bar Graph
• Pie Chart
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Frequency Distribution
• A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several non overlapping classes.
• The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.
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Bar Graph
• A bar graph is a graphical device for depicting qualitative data.
• Using a bar of fixed width drawn above each class label, we extend the height appropriately.
• The bars are separated to emphasize the fact that each class is a separate category.
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Bar Graph
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The simple bar diagram
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Compound bar diagram
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Polybar diagram
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Pie Chart
• The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
• First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
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Pie Chart
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Pie graphs
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Quantitative Data
• Frequency Distribution
• Histogram
• Cumulative Distributions
• Ogive
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Frequency Distribution
• Selecting Number of Classes Use between 5 and 20 classes. Data sets with a larger number of elements usually require a larger number of classes. Smaller data sets usually require fewer
classes. • Selecting Width of Classes Use classes of equal
width. Approximate Class Width = Largest Data Value Smallest Data Value
Number of Classes Largest Data Value Smallest Data Value
Number of Classes
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Histogram
• Another common graphical presentation of quantitative data is a histogram.
• A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency
• Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.
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Histogram
PartsCost ($) PartsCost ($)
2244
66
88
1010
1212
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1616
1818
Freq
uenc
yFr
eque
ncy
50 60 70 80 90 100 11050 60 70 80 90 100 110
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Ogive
• An ogive is a graph of a cumulative distribution.The data values are shown on the horizontal axis.
• Shown on the vertical axis are the:
cumulative frequencies,
• The frequency (one of the above) of each class is plotted as a point.
• The plotted points are connected by straight lines.
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Ogive with Cumulative Frequencies
PartsPartsCost ($)Cost ($) PartsPartsCost ($)Cost ($)
2020
4040
6060
8080
100100
Cum
ulati
ve P
erce
nt F
requ
ency
Cum
ulati
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erce
nt F
requ
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Cum
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erce
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requ
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Cum
ulati
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erce
nt F
requ
ency
50 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 110
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Inferential statistics are designed to allow inference from a statistic measured on sample of cases to a population parameter. Used to test hypotheses about the population as a whole.
Types of Statistics
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Inferential Statistics
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Statistical Tests
• Parametric tests
Continuous data normally distributed
• Non-parametric tests
Continuous data not normally distributed
Categorical or Ordinal data
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Level of Measurement
Statistic
Nominal Mode What is the most frequent value?
Ordinal MedianWhat is the middle score? (50% above and 50% below)
Interval/Ratio MeanWhat is the average? (Sum of all scores divided by the number of scores)
Measures of Central Tendency
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Example of Central Tendency
15,20,21,20,36,15,25,15
15,15,15,20,20,21,25,36
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Example of Mode
Statistics
RACE Race of Respondent1500
0
1
Valid
Missing
N
Mode
Race of Respondent
Race of Respondent
otherblackw hite
Fre
quen
cy
1400
1200
1000
800
600
400
200
0
RACE Race of Respondent
1257 83.8
168 11.2
75 5.0
1500 100.0
1 white
2 black
3 other
Total
Frequency Percent
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Example of Median
Statistics
EDUC Education level24
0
6.00
Valid
Missing
N
Median
24N =
Education level
10
9
8
7
6
5
4
3
EDUC Education level
1 4.2 4.2
6 25.0 29.2
6 25.0 54.2
3 12.5 66.7
4 16.7 83.3
4 16.7 100.0
24 100.0
4 Some high school
5 Completed high school
6 Some college
7 Completed college
8 Some graduate work
9 A graduate degree
Total
Frequency PercentCumulative
Percent
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Age of Respondent
908580757065605550454035302520
Age of Respondent200
100
0
Std. Dev = 17.42
Mean = 46
N = 1495.00
Example of Mean
MEAN
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Mean versus Median
• Large sample values tend to inflate the mean. This will happen if the histogram of the data is right-skewed.
• The median is not influenced by large sample values and is a better measure of centrality if the distribution is skewed.
• Note if mean=median=mode then the data are said to be symmetrical
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Measures of Dispersion
• Measures of dispersion characterise how spread out the distribution is, i.e., how variable the data are.
• Commonly used measures of dispersion include:1. Range2. Variance & Standard deviation3. Coefficient of Variation (or relative standard
deviation)4. Inter-quartile range
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Level of Measurement
Statistic
NominalNumber of categories
How many different values are there?
Ordinal RangeWhat are the highest and lowest values?
Interval/RatioStandard Deviation
What is the average deviation from the mean?
Measures of Variation
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Curves of Distribution
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Normal Distribution
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Normal Curve
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Example: Number of categories
Race of Respondent
Race of Respondent
otherblackw hite
Freq
uenc
y
1400
1200
1000
800
600
400
200
0
RACE Race of Respondent
1257 83.8
168 11.2
75 5.0
1500 100.0
1 white
2 black
3 other
Total
Frequency Percent
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Example of Range
24N =
Education level
10
9
8
7
6
5
4
3
EDUC Education level
1 4.2 4.2
6 25.0 29.2
6 25.0 54.2
3 12.5 66.7
4 16.7 83.3
4 16.7 100.0
24 100.0
4 Some high school
5 Completed high school
6 Some college
7 Completed college
8 Some graduate work
9 A graduate degree
Total
Frequency PercentCumulative
Percent
Statistics
EDUC Education level24
0
6.00
5
4
9
Valid
Missing
N
Median
Range
Minimum
Maximum
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Example of Standard Deviation
Age of Respondent
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
Age of Respondent
Fre
qu
en
cy
200
100
0
Std. Dev = 17.42
Mean = 46
N = 1495.00
MEAN +1 SD-1 SD
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Measures of Relationships
Level of Measurement
Statistic
Nominal Phi statistic ()
OrdinalSpearman rho () correlation
Interval/Ratio Pearson correlation (r)
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Correlation
• Assesses the linear relationship between two variablesExample: height and weight
• Strength of the association is described by a correlation coefficient- r• r = 0 - .2 low, probably meaningless• r = .2 - .4 low, possible importance• r = .4 - .6 moderate correlation• r = .6 - .8 high correlation• r = .8 - 1 very high correlation
• Can be positive or negative• Pearson’s, Spearman correlation coefficient • Tells nothing about causation
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Examples of Some Commonly Used Statistical Tests
Level of Measurement
Number of groups Nominal Interval/Ratio
1 group 2 testt-test of sample mean vs. known population value
2 independent groups2 test
Independent samples t-test
2 dependent groupsMcNemar test
Paired t-test
>2 independent groups 2 test ANOVA
>2 dependent groupsCochran Q test
Repeated measures ANOVA
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Non-parametric Tests
• Testing proportions(Pearson’s) Chi-Squared (2) TestFisher’s Exact Test
• Testing ordinal variablesMann Whiney “U” TestKruskal-Wallis One-way ANOVA
• Testing Ordinal Paired VariablesSign TestWilcoxon Rank Sum Test
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Use of non-parametric tests
• Use for categorical, ordinal or non-normally distributed continuous data
• May check both parametric and non-parametric tests to check for congruity
• Most non-parametric tests are based on ranks or other non- value related methods
• Interpretation:Is the P value significant?
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Chi-Squared (2) Test
• Used to compare observed proportions of an event compared to expected.
• Used with nominal data (better/ worse; dead/alive)
• If there is a substantial difference between observed and expected, then it is likely that the null hypothesis is rejected.
• Often presented graphically as a 2 X 2 Table
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Analysis of Variance
• Used to determine if two or more samples are from the same population- the null hypothesis.If two samples, is the same as the T test.
Usually used for 3 or more samples.
• If it appears they are not from same population, can’t tell which sample is different.Would need to do pair-wise tests.
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Tests of Hypotheses – Tests of Significance
Designed experiment - only two explanations for a negative answer, difference is due to the applied treatments or a chance effect
Survey is silent in distinguishing between various possible causes for the difference,
merely noting that it exists.
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Tests of Hypotheses - Tests of Significance
Survey: Are the observed differences between groups compatible with a view that there are no differences between the populations from which the samples of values are drawn?
Designed experiments: Are observed differences between treatment means compatible with a view that there are no differences between treatments?
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Standard Error
• Standard error of the meanStandard deviation / square root of (sample size)
• (if sample greater than 60)
• Standard error of the proportionSquare root of (proportion X 1 - proportion) / n)
• Important: dependent on sample sizeLarger the sample, the smaller the standard error.
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Errors
• Type I error
Claiming a difference between two
samples when in fact there is none.Also called the error.Typically 0.05 is used
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Errors
• Type II errorClaiming there is no difference between
two samples when in fact there is.Also called a error.The probability of not making a Type II
error is 1 - ,Hidden error because can’t be detected
without a proper analysis
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Hypothesis Testing Decision Chart
Null Hypothesis (H0 ) is true Alternative Hypothesis (H1) is true
Reject (H0 )
Type I error()
typically .05 or .01
Correct decision
(Power = 1 - )
typically .80
Don’t reject (H0 )
Correct decision
(1 - )
typically .95 or .99
Type II error()
typically .20
Decision
Reality
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Males and females are asked a question that is measured on a five-point Likert scale:
Do males and females differ in their response to this question?
1 Strongly agree2 Agree3 Neither agree nor disagree4 Disagree5 Strongly disagree
To what extent do you feel that regular exercise contributes to your overall health?
Difference between two group means: The independent
samples t-test
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Comparison of 2 Sample Means
• Student’s T testAssumes normally distributed continuous
data.T value = difference between means
standard error of difference
T value then looked up in Table to determine significance
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females
1 2 3 4 5
111
males
meanmales=2.5
meanfemales=3.2
25 males and 25 females answered our question. Here
is how they responded:
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Group Statistics
25 2.56 1.158 .232
25 3.24 1.012 .202
GENDER1 male
2 female
EXERCISEN Mean Std. Deviation
Std. ErrorMean
Independent Samples Test
-2.212 48 .032 -.68EXERCISEt df Sig. (2-tailed)
MeanDifference
t-test for Equality of Means
The t-test reveals a significant difference between males & females:
Student t test
