Week 1

Part 1

Introduction to the Course

The Nature of Data

Why Statistics?

• Evidence-based practice!

• Research provides evidence for changes in nursing/medical practice– Away from “that’s the way it has always been

done”

Integral to Research

• Question (hypothesis)

• Design

• Data collection

• Analysis

• Answer to question– And often more questions asked!

Data

• Data: – Factual information, especially information

organized for analysis or used to reason or make decisions; a fact or proposition used to draw a conclusion or make a decision

• The American Heritage® Dictionary of the English Language, Fourth EditionCopyright © 2000 by Houghton Mifflin Company.

– Datum: Singular of data• an item of factual information derived from

measurement or research

Two Types of Data

• Qualitative– Non-numeric or narrative information

• Example: transcripts of interviews• Maybe “scored” to be made quantitative

• Quantitative– Numeric or quantifiable information

• Example: weights of kindergartners

Variable

• A quantity capable of assuming a set of values

• A characteristic or attribute of a person, object, etc that varies within a population under study

• Examples:– Body temperature, BP, DOB, ABG, weight

Independent and Dependent

• Independent– The variable assumed to influence the

outcome• It is independent of the outcome

– In research, the manipulated variable

• Dependent– The outcome variable of interest– In research, value assumed to be dependent

on the independent variable (by hypothesis)

Independent and Dependent

• Examples:– What is the effect of smoking on the incidence

of lung cancer?– Does high fiber diet reduce the risk of colon

cancer?– Does AZT help prevent maternal transmission

of HIV?

Discrete vs Continuous

• Discrete variable: has a finite number of values between two points

• Continuous variable: has, in theory, an infinite number of values between two points

Discrete vs Continuous

• Examples:– Number of children– Body temperature– Hospital readmissions– Chemotherapy sessions– Body weight– DOB

Measurement

• The assignment of numbers to objects according to specified rules to characterize quantities of some attribute

Measurement Rules

• Common/familiar/accepted– Temperature, weight, height

• Researcher designed– Particularly for new materials/ideas

• Coding– The process of transforming raw data into

standardized form for processing and analysis

Advantages of Measurement

• Objectivity– Objective measure can be independently

verified by other researchers

• Precision– Quantitative measures allow for reasonable

precision

• Communication– Facilitates communication of data and

research

Levels of Measurement/Types of Variables

• Nominal

• Ordinal

• Interval

• Ratio

Nominal Measurement/Variable

• Nominal = Named

• Lowest level

• Assignment of characteristics into categories– Simply putting into boxes with no meaning of

where the boxes fall in a line

• Examples– Gender, marital status

Ordinal Measurement/Variable

• Ordinal=Order

• Next in the hierarchy of measurement

• Involves rank order of variable along some dimension

• Examples– School grades– Clinical nursing levels

Interval Measurement/Variable

• Interval=equal distances

• Attribute is rank-ordered on a scale that has equal distances between points on that scale

• Examples– Temperature

Ratio Measurement/Variables

• Equal distances between score units and which has a true, meaningful zero point– A true ratio can be calculated

• The highest level of measurement

• Examples– Weight– Pulse

Why care about type of measurement/variable?

• Statistical tests are/have been developed to work and provide meaningful analysis for specific types of measurement and variable

• The tests you choose to run should be based, in part, on the type of variables with which you work

Which measurement?

• A single variable may be measurable using different types of measurement

• Rule of Thumb: use the highest level of measurement possible– Higher levels provide more information– Higher levels can be analyzed with more

powerful statistical tools

Data Analysis

• Data starts out “raw”– unanalyzed

• Processing– Coding, if appropriate– Data entry

• Into database or matrix

– Cleaning• Finding and correcting (if possible) errors in entry

and coding

– Analysis

Sample vs Population

• Sample– A subset of a population – Ideally selected to be representative of the

population

• Population– The entire set of individuals (objects, units,

etc) having common characteristics

Two Types of Statistics

• Descriptive– Used to describe and summarize data set– Allows us to describe, compare, determine a

relationship– Usually straightforward - %, averages, etc

• Inferential– Permit us to infer whether a relationship

observed in a sample is likely to occur in the population of concern

– Are relationships “real”?

Uses of Inferential Statistics

• Draw conclusions about a single variable in a population

• Evaluate relationships between variables in populations

• Are the relationships “real”?

Inferential Stats: Relationships

• Existence– Is there a relationship between X and Y?

• Magnitude– How strong is the relationship between X and

Y?

• Nature– What type of relationship is there between X

and Y?

Number of variables…

• “Univariate”– One variable being described

• “Bivariate”– Two variables being compared

• NOTE: in epidemiology, this is also known as “univariate”

• Mulitvariate– More than two variables being compared

• Different statistical tests for each

Purposes of Data Analysis

• In research all usually get done to some extent– Clean data– Sample description– Assessment of bias– Evaluation of tools used to collect data– Evaluation of need for data transformations– Address the research question

Describing the Data Set

• Organize the data

• Examine the patterns of distribution

• Describe patterns of distribution

• Asses the variability of the data

Simplest Distribution: The Frequency Distribution

• Lists categories of scores or values as well as counts of the number of each score or value– List and tally– By computer

• Enter data• Run “frequency”

Two Kinds of Frequency

• Absolute– Number of times a score occurs– Symbol: f

• Relative– Proportion of times a score occurs– Most commonly percent

• % = (f/N) X 100– f=frequency, N=sum of all frequencies

Frequency Example: Blood Pressure (mm Hg) Readings in

an Anti-Hypertensive Trial – Raw Data

166 160 166 162 168 148 164 174 164 188

176 170 166 172 168 172 150 190 164 150

164 146 178 154 166 148 156 164 180 166

172 170 180 156 162 176 184 166 174 158

186 158 166 170 168 178 178 154 166 152

168 160 168 166 152 160 170 146 186 176

n=60

Frequency Distribution

x f x f x f

146 1 162 4 178 2

148 2 164 5 180 2

150 2 166 9 182 2

152 2 168 5 184 1

154 2 170 4 186 2

156 2 172 3 188 1

158 2 174 2 190 1

160 3 176 2 192 0n=60

Relative Frequency (rf) Distribution

x rf x rf x rf146 0.03 162 0.03 178 0.05

148 0.03 164 0.08 180 0.03

150 0.03 166 0.15 182

152 0.03 168 0.08 184 0.02

154 0.03 170 0.07 186 0.03

156 0.03 172 0.05 188 0.02

158 0.03 174 0.03 190 0.01

160 0.05 176 0.05 192

n=60

Cumulative Relative Frequency (Cf) Distribution

x Cf x Cf x Cf146 0.03 162 0.32 178 0.88

148 0.07 164 0.40 180 0.91

150 0.10 166 0.55 182 0.91

152 0.13 168 0.63 184 0.93

154 0.17 170 0.70 186 0.96

156 0.20 172 0.75 188 0.98

158 0.23 174 0.78 190 1

160 0.28 176 0.83 192

n=60

Grouped Frequency Distribution

• Values are grouped into intervals– Class intervals are all the same size– Class intervals are mutually exclusive– Useful when data is dispersed

• Or there are restrictions on “small cell size”– For example: HIV/AIDS reporting

– Loss of information with grouping• Anytime one moves from the individual level to

group level

Grouped Frequency Distribution

Interval f

<150mm Hg 4

150-158mm Hg 10

160-168mm Hg 24

170-178mm Hg 15

180-188mm Hg 6

≥188mm Hg 1n=60

Displaying Data

• Tables

• Bar graphs

• Pie charts

• Histograms

• Frequency Polygons– aka Line charts/graphs

Bar Graph

• Used primarily for nominal and ordinal data

• Values across the X axis

• Frequencies along the Y axis

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1

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3

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9

10

146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190

mm Hg

num

ber o

f sub

ject

s

Bar Graph of Hypertension Data

(Generated in Excel)

Histogram

• Like a bar graph

• Used for continuous (interval or ratio) data– Rarely seen even for interval or ratio data– Not offered as an option in Excel

• Bars touch

• May use grouped data

Bar Chart and Histogram

150 160 170 180 190

BP1

0

2

4

6

8

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12

14

Fre

qu

en

cy

Mean = 166.43Std. Dev. = 10.692N = 60

146

148

150

152

154

156

158

160

162

164

166

168

170

172

174

176

178

180

184

186

188

190

BP1

0

2

4

6

8

10

Cou

nt

(Generated in SPSS)

Frequency Polygon (aka Line Graph)

• Used for interval and ratio data

• X and Y axes the same as for bar charts

• Marker placed at intersection of the value and frequency for a series of values

• Markers then connected with a line

Frequency Polygon of Hypertension Data

0

1

2

3

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5

6

7

8

9

10

146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190

mm Hg

nu

mb

er o

f su

bje

cts

Effective Graphical Display

• Should accurately represent data

• Should be easily understood– Not too busy or complicated

• Should stand alone– Ideal and rare

Distribution Shapes – 5 Basics

• Modality

• Symmetry and Skewness

• Kurtosis

• Central Tendency

• Variability

Modality – Basic Shape

• Peaks or high points in the data

• May have one or multiple peaks– Unimodal = 1 peak– Bimodal = 2 peaks– Multimodal = multiple

peaks

Symmetry

• Symmetrical– If you draw a line through the center it

produces mirror images– In real life: approximately the same

distribution on either side of the center line

• Asymmetrical– Distribution is lopsided or skewed

Asymmetric Distribution: Skewness

• Affected by outliers• Positive

– The “tail” points to the right (positive direction)

• Negative– The “tail” points to the

left (negative direction)

Kurtosis

• Assumes symmetric distribution

• Refers to how pointy the peak of the distribution is– How concentrated in the

middle of the distribution

• Platykurtic– Low, flattened peak

• Leptokurtotic– High narrow peak

The Normal Distribution

• Unimodal

• Symetrical

• Peak is neither high nor flat

• “Bell-shaped curve”

• The ideal distribution– And therefore “normal”

Looking at Frequency Distributions

• Learn about the data set

• Clean the data

• Identify missing values

• Test assumptions– About the distribution

• Answer research questions– About the distribution

Quartiles

• Calculated by dividing data into quarters– The median is the 2nd quartile

• Quartile 1 is the point at which 25% of values are below and 75% of values are above

• Quartile 3 is the point at which 75% of values are below and 25% of values are above

Part 2

Describing and Displaying Data

Measures of Central Tendency

Univariate Statistics

Measures of Central Tendency

• Tells you about the area of the distribution where the bulk of values fall

• Measures include:– Mean– Median– Mode

Mode

• The value that occurs most often

• Limitations– Data may be multimodal– Mode can vary from one sample to another in

the same population• Considered unstable

Median (Mdn)

• Point that divides the distribution in half

• Corresponds to the 50th percentile

• 50% will be below the median and 50% above it

• If the number of scores is odd– Median is the number exactly in the middle

• If the number of scores is even– Median is the average of the 2 middle numbers

Median (Mdn)

• Measures the location of the middle of the distribution

• Not sensitive to actual numerical values– Not affected by outliers

Mean

• Most common measure of central tendency• Most stable, provides the most accurate

estimate– Assuming a normal distribution

• Calculated by adding all values and dividing by the number of cases– aka average– Best understood by the general public

Mean

• Affected by each value in the distribution

• Intended for interval or ratio data– In some designs can be used for ordinal

• The sum of the deviation scores from the mean always equals 0

• Abbreviated x for samples– X for Population

Mean, Median, and Mode

• Mean is preferred in a normal distribution– Extreme scores or outliers can result in a

mean that doesn’t reflect central tendency• Skewed data

• With skewed data, or extreme outliers use median– Example: Median home price

x, Mdn, Mode – Hypertension Study

Sum of values (x) = Σ(x) = 9,989 Number of cases = n = 60

x f x f x f146 1 162 4 178 2

148 2 164 5 180 2

150 2 166 9 182 2

152 2 168 5 184 1

154 2 170 4 186 2

156 2 172 3 188 1

158 2 174 2 190 1

160 3 176 2 192 0

Mode = 166Mdn = 166

Mean = 9991/60 = 166.5

Quickly Assessing Distribution

• If the mean, median, and mode are similar– Approximately normally distributed

• If the median>mean– Negatively skewed

• If the median<mean– Positively skewed

• The mean is pulled in the direction of the skew

Effect of Skew

ModeMode MedianMedian

MeanMean

Mode Mode

MedianMedian

MeanMean

The mean is pulled in the directionThe mean is pulled in the directionof the skewof the skew

Variability

• Refers to how spread out the scores are in a distribution

• Two distributions with the same mean can differ greatly in variability– Homogeneous: values are similar– Heterogeneous: values with more variability

• Measures:– Range/Semiquartile Range– Variance– Standard Deviation

Variability – Spread or Dispersion

0

1

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8

9

10

146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190

mm Hg

nu

mb

er o

f su

bje

cts

Range

• Simplest of the measures of variability

• Difference between the lowest and highest values in a distribution– 190-146 = 44

• Sometimes reported as a minimum and maximum value– Range 146-190

Range

• Limitations– Based on only 2 values, highest and lowest

• Can be unstable when multiple samples are taken from the same population

• Doesn’t tell you anything about what is happening in the middle

– As the sample size increases, range is likely to increase

• Greater chance of outlier

Standard Deviation - s, SD, Std Dev

• A measure of how far values vary from the mean of a given sample– Tells you the average deviation – How much the scores deviate from the mean

• Most widely used measure of variability

• Takes into consideration every score in the distribution

Standard Deviation

Standard deviation = s = Σ (X-mean)2

N-1

X X-mean (X-mean)2

7 7-4=3 95 5-4=1 14 4-4=0 03 3-4= -1 11 1-4= -3 9

N=5 Σ 20Mean=4

s= 205-1

= √ 5 = 2.24

Standard Deviation

• Taking the square root returns the value of the standard deviation to the original scale

• The lower the standard deviation, the better measure the mean is as a summary of the data– The less variability there is among the scores

Variance

• Simply s2

• The standard deviation calculation before the square root is taken and is equal to:

Σ (X-mean)2 /N-1

Standard Deviation - uses

• Useful when looking at a single score in relation to a distribution

• Normal Distribution– There are about 3 SD above and below the

mean– A fixed percent of scores lie within each SD:

• 68% within 1 SD above and below the mean• 95% within 2 SD above and below the mean• 99.7% within 3 SD above and below the mean

Normal Distribution with SD = 15and mean = 100

2.5% 13.5% 34% 2.5%13.5%34%

68%

95%

1008570 115 130

Standard Scores

• Scores that represent relative distance from mean. Measures of position.

Z= X-X/SD

• Raw score minus mean divided by SD: gives score in SD units

• Z score is # of SDs a given value of ‘X’ is away from mean. Z score of 1 is 1 SD above mean.

• Z Distribution has mean = 0 and SD = 1

Standard Scores – Z scores

• Allow for the standardization (in SD units) of values in a distribution relative to the mean

• Standard Score Z = (x-x)/SD

• Number of SD a given value of x is from the mean– Z score of 1 is 1 SD above the mean

• Z distribution has mean=0 and SD=1

55 60 65 85 100 115

-1 0 +1

Score - Mean --------------------- SD

Z Distribution

Score - Mean --------------------- SD

Z Score = Z Score =

Normal Z Distribution (Mean = 0, SD = 1)

2.5% 13.5% 34% 2.5%13.5%34%

68%

95%

0-1-2 +1 +2

Normal Distribution/Z Scores

• The entire percent under the curve is 100%– Probability of being somewhere under the

curve is 100%

• Most values will lie in the middle

• Out at the ends we become less sure– Is a value out at 1% really representative?

Using Normal Distributions/Z Scores

• Transformation– The z score can be transformed to reset the

mean and SD• Transformed Z = 10(Z) + 50

– Now mean = 50 and SD=10

• P-value– Likelihood of a given value falling at a

particular point on the curve– We will come back to this

Using Normal Distributions/Z Scores

• Z score can tell you the probability of a value falling into a given area of the curve– Get z score– Match to %

• Z= 2 corresponds to 95%

– Gives the probability of the value being the true mean

– Z-score tables

Parameters vs Statistics

• Parameters– Computed for populations– Greek symbols used

• μ =mean, σ = std dev

• Statistics– Computed for samples– English symbols used

• X =mean, s = std dev

Computers and Measures of Central Tendency

• Statistical software is in widespread use but…

• The operator (you) must be aware of levels of measurement etc– The computer doesn’t know– Have to choose the right method for type of

data

“Bivariate” Statistics

“Bivariate” Statistics

• Used to describe the relationship between 2 variables (bi-variate)– 2 nominal variables– 1 nominal, 1 ratio/interval– 2 ratio/interval

Crosstabulation

• Results in a contingency table– 2 dimensional frequency distribution

• The simplest: 2X2– 2 nominal or ordinal variables

• One heading columns• One heading rows

Crosstabulation - Example

High School

College total

<$30,000 64 19 83

≥$30,000 36 81 117

total 100 100 200

Comparison of Group Means

• Nominal & Interval or Ratio Variable

• IV: nominal or ordinal– Sex, ethnicity, age group etc

• DV: interval or ratio– Heart rate, BP, weight etc

• Means and SD calculated for each category of the IV

• NOTE: NO INFERENCE is made about significance of difference between categories

Comparison of Group Means

education n mean SD Min Max

High School 100 24,657 2,598 10,103 75,362

College 100 36,431 7,912 15,256 126,754

total 200 31,989 6,110 10,103 126,754

Correlation

• A linear relationship between 2 variables– Interval or ratio variables

• Can be plotted and displayed graphically– Scatter plot

• Can be calculated statistically– Correlation coefficient– r

Scatterplot

• Values for one variable on X axis

• Values for the other variable on Y axis

• Data plotted for each subject/case

• Examine the plot for pattern– Data arrayed closely together indicates strong

correlation

Positive Correlation

• As one variable increases in value, so does the other

• On the plot:– Diagonal line upwards

and to the right

• Example:– Age and BP

Negative Correlation

• As one value increases the other decreases

• On the plot:– Diagonal line down

and to the right

• Example:– Age and bone density

Scattered Scatterplots

• Indicate little or no relationship between variables

• Can be dispersed or concentrated

Non-linear Scatterplots

• There is a relationship but…

• Some relationships are not linear….

• May be curved– S– U– Up then flat– others

Outliers on Scatterplots

• Scatterplots can also help identify where outliers are

Correlation Coefficient - r

• Statistical measure used to– Determine if a relationship exists between two

variables– Test a hypothesis about that relationship

• Allows us to make a mathematical statement about the relationship– Do the variables vary together?

• AKA Pearson Correlation Coefficient

Correlation Coefficient - Assumptions

• Sample must accurately representative

• The distributions must be approximately normal

• Each value of X must have a corresponding value of Y– If many have X value but not Y value, analysis

will be strongly biases

Correlation

r = n(∑xy) – (∑x)(∑y)

[n(∑x2) - (∑x)2 ] [n(∑y2) - (∑y)2 ]

= cov(X,Y)

var(X) x var(Y)

Correlation Examplex y xy x2 y2

8 -2 -16 64 4

4 2 8 16 4

5 1 5 25 1

-1 6 -6 1 36

1 4 4 1 16

2 3 6 4 9

6 -1 -6 36 1

x=25 y=13 y=-5 x2=147 y2=71

r= 7(-5) – (25)(13)

[7(147)-(25)2] x [7(71)-(13)2]

= -0.989

Correlation Coefficient

• Range– -1 to 1

• Positive correlation– 0 to1

• Negative correlation– -1 to 0

• The closer to each of these the stronger the correlation– -0.9: strong negative– -0.2: weak negative– 0: none– 0.2: weak positive– 0.9 strong positive

Correlation Coefficient – Significance

• Depends on number of pairs

• Varies for each r– r of 0.3 may be significant for n=1500 but not

for n=40

• Also depends on variance (SD)– Greater the variance, less significance

• Generally:– 0.60 or –0.60 is strong for medical variables

• Manufacturing requires 0.90 or greater

The Scatterplot and r

• ALWAYS look at the scatterplot along with r

• Each of these plots has r=0.70

Correlation

• The square of the correlation coefficient, R2, indicates the variability in one variable that can be explained by the other– Example: age and BP

• R2 = 0.49 (r=0.70)• 49% of the variation in BP is explained by age

– aka Coefficient of Determination

• Correlation does NOT imply causation