© 2006

Dr Rotimi Adigun

Recommended text

High Yield Biostatistics, Epidemiology and Public Health ,Michael Glaser, Fourth edition.

Pre-Test Preventive Medicine and Public Health, Sylvie Ratelle(Chapter 1) for practice questions.

I. Descriptive statistics Populations, samples

and Elements Probability Types of Data Frequency Distribution Measures of Central

Tendency Measures of Variability Z scores

II. Inferential Statistics Statistics and

Parameters Estimating mean T-scores Hypothesis testing Steps of hypothesis

testing Z-tests The meaning of

statistical significance Type I and II errors Power Differences between

groups

III. Correlation and

Predictive Techniques

Correlation Survival analysis

IV.

Research methods Sampling techniques Assessing the

Evidence Hierarchy of

evidence Systematic review Clinical Decision

making

© 200604/21/23

Descriptive Statistics,Population, Samples, Elements

Types of Data, Measures of Central tendency, Measures of Variability,

Z-scores.

A way to summarize data from a sample or a population

DSs illustrate the shape, central tendency, and variability of a set of data The shape of data has to do with the

frequencies of the values of observations

DSs= descriptive statistics.

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Population A population is the group from which a sample is drawn e.g., automobile crash victims in an

emergency room ,Student scores on block I exams at Windor,

In research, it is not practical to include all members of a population

Thus, a sample (a subset of a population) is taken

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Elements=A single observation—such as one students score—is an element, denoted by X.

The number of elements in a population is denoted by N

The number of elements in a sample is denoted by n.

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Data Measurements or observations of a variable

Variable A characteristic that is observed or

manipulated Can take on different values

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Independent variables Precede dependent variables in time Are often manipulated by the researcher The treatment or intervention that is used in a

study Dependent variables

What is measured as an outcome in a study Values depend on the independent variable

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Independent and Dependent variables e.g.

-You study the effect of different drugs on colon cancer, the drugs, dosage, timing would be independent variables, the effect of the different drugs, dosage ,and timing on cancer would be the dependent variable.

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We compare the effect of stimulants on memory and retention.

Independent variables would be the different stimulants and dosages..dependent variables would be the different outcomes..(improved retention, decreased retention or no effect)

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Mathematically on a graph or equation where Y depends on the value of X,

Y is a function of X or f(x)

Independent variable is usually plotted on the X axis while the dependent variable is plotted on the Y axis.

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Parameters Summary data from a population

Statistics Summary data from a sample

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Central tendency describes the location of the middle of the data

Variability is the extent values are spread above and below the middle values a.k.a., Dispersion

DSs can be distinguished from inferential statistics DSs are not capable of testing hypotheses

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Mean (a.k.a., average) The most commonly used DS

To calculate the mean Add all values of a series of numbers and then

divided by the total number of elements

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Mean of a sample

Mean of a population

(X bar) refers to the mean of a sample and refers to the mean of a population

X is a command that adds all of the X values n is the total number of values in the series of a

sample and N is the same for a population

X μ

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N

X

n

XX

Mode The most frequently

occurring value in a series

The modal value is the highest bar in a histogram

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ModeMode

Median The value that divides a series of values in

half when they are all listed in order When there are an odd number of values

The median is the middle value When there are an even number of values

Count from each end of the series toward the middle and then average the 2 middle values

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Each of the three methods of measuring central tendency has certain advantages and disadvantages

Which method should be used? It depends on the type of data that is being

analyzed e.g., categorical, continuous, and the level of

measurement that is involved

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There are 4 levels of measurement Nominal, ordinal, interval, and ratio

1. Nominal Data are coded by a number, name, or letter

that is assigned to a category or group No ordering Examples

Gender (e.g., male, female) Race Treatment preference (e.g., Surgery,

Radiotherapy, Hormone, Chemotherapy)

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2. Ordinal Is similar to nominal because the

measurements involve categories However, the categories are ordered by rank Examples

Pain level (e.g., mild, moderate, severe) Military rank (e.g., lieutenant, captain, major,

colonel, general) Opinion- (Agree, strongly agree) Severity – Mild, moderate, severe (dysplasia)

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Ordinal values only describe order, not quantity Thus, severe pain is not the same as 2 times

mild pain The only mathematical operations

allowed for nominal and ordinal data are counting of categories e.g., 25 males and 30 females

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3. Interval Measurements are ordered (like ordinal data) Have equal intervals Does not have a true zero Examples

The Fahrenheit scale, where 0° does not correspond to an absence of heat (no true zero)

In contrast to Kelvin, which does have a true zero

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4. Ratio Measurements have equal intervals There is a true zero Ratio is the most advanced level of

measurement, which can handle most types of mathematical operations (including multiplication and division which are absent in interval scales)

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Ratio examples Range of motion

No movement corresponds to zero degrees The interval between 10 and 20 degrees is the

same as between 40 and 50 degrees Lifting capacity

A person who is unable to lift scores zero A person who lifts 30 kg can lift twice as much as

one who lifts 15 kg

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NOIR is a mnemonic to help remember the names and order of the levels of measurement Nominal

OrdinalIntervalRatio

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Measurement scalePermissible mathematic

operationsBest measure ofcentral tendency

Nominal Counting Mode

OrdinalGreater or less than

operationsMedian

Interval Addition and subtractionSymmetrical – Mean

Skewed – Median

RatioAddition, subtraction,

multiplication and division Symmetrical – Mean

Skewed – Median

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Histograms of frequency distributions have shape

Distributions are often symmetrical with most scores falling in the middle and fewer toward the extremes

Most biological data are symmetrically distributed and form a normal curve (a.k.a, bell-shaped curve)

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Line depicting the shape of the data

Line depicting the shape of the data

The area under a normal curve has a normal distribution (a.k.a., Gaussian distribution)

Properties of a normal distribution It is symmetric about its mean The highest point is at its mean The height of the curve decreases as one

moves away from the mean in either direction, approaching, but never reaching zero

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MeanMean

A normal distribution is symmetric about its meanA normal distribution is symmetric about its mean

As one moves away from the mean in either direction the height of the curve decreases, approaching, but never reaching zero

As one moves away from the mean in either direction the height of the curve decreases, approaching, but never reaching zero

The highest point of the overlying normal curve is at the mean

The highest point of the overlying normal curve is at the mean

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Mean = Median = ModeMean = Median = Mode

The data are not distributed symmetrically in skewed distributions Consequently, the mean, median, and mode

are not equal and are in different positions Scores are clustered at one end of the

distribution A small number of extreme values are located

in the limits of the opposite end

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Skew is always toward the direction of the longer tail(not the hump) Positive if skewed to the right Negative if to the left

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The mean is shifted the most

Because the mean is shifted so much, it is not the best estimate of the average score for skewed distributions

The median is a better estimate of the center of skewed distributions It will be the central point of any distribution 50% of the values are above and 50% below

the median

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Mean,Median and Mode

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Midrange Smallest observation + Largest

observation2

Mode the value which occurs with the

greatest frequency i.e. the most common value

Summary statistics

Median the observation which lies in the

middle of the ordered observation.

Arithmetic mean (mean)Sum of all observationsNumber of observations

Summary statistics

The mean represents the average of a group of scores, with some of the scores being above the mean and some below This range of scores is referred to as variability

or spread Range Variance Standard deviation Semi-interquartile range Coefficient of variation “Standard error”

SD is the average amount of spread in a distribution of scores

The next slide is a group of 10 patients whose mean age is 40 years Some are older than 40 and some younger

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Ages are spread out along an X axis

Ages are spread out along an X axis

The amount ages are spread out is known as dispersion or spread

The amount ages are spread out is known as dispersion or spread

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Adding deviations always equals zero

Adding deviations always equals zero

Etc.

To find the average, one would normally total the scores above and below the mean, add them together, and then divide by the number of values

However, the total always equals zero Values must first be squared, which cancels

the negative signs

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Symbol for SD of a sample for a population

Symbol for SD of a sample for a population

S2 is not in the same units (age), but SD is

S2 is not in the same units (age), but SD is

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7 7

7 7 7

7

7 8

7 7 7

6

3 2

7 8 13

9

Mean = 7SD=0

Mean = 7SD=0.63

Mean = 7SD=4.04

About 68.3% of the area under a normal curve is within one standard deviation (SD) of the mean

About 95.5% is within two SDs About 99.7% is within three SDs

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The number of SDs that a specific score is above or below the mean in a distribution

Raw scores can be converted to z-scores by subtracting the mean from the raw score then dividing the difference by the SD

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X

z

If the element lies above the mean, it will have a positive z score;

if it lies below the mean, it will have a negative z score.

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Standardization The process of converting raw to z-scores The resulting distribution of z-scores will

always have a mean of zero, a SD of one, and an area under the curve equal to one

The proportion of scores that are higher or lower than a specific z-score can be determined by referring to a z-table

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Refer to a z-tableto find proportionunder the curve

Refer to a z-tableto find proportionunder the curve

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Partial z-table (to z = 1.5) showing proportions of the area under a normal curve for different values of z.

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94410.93320.9332

Corresponds to the area under the curve in black

Corresponds to the area under the curve in black

Tables of z scores state what proportion of any normal distribution lies above or below any given z scores, not just z scores of ±1, 2, or 3.

Z-score tables can be used to - Determine proportion of distribution with a

certain score(e.g finding the proportion of the class with a score of 65% on an exam)

-Find scores that divide the distribution into certain proportions(for example finding what scores separates the top 5% of the class from the remaining 95%)

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Allows us to specify the probability that a randomly picked element will lie above or below a particular score.

For example, if we know that 5% of the population has a heart rate above 90 beats/min, then the probability of one randomly selected person from this population having a heart rate above 86.5 beats/min will be 5%.

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What is the probability that a randomly picked person would have a heart rate of less than 50 beats per minute in a population with S.D of 10 and mean heart beat of 70z?

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