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1© 2006
Biostatistics Basics
An introduction to an expansive and complex field
Evidence-based Chiropractic 2 © 2006
Common statistical terms
• Data – Measurements or observations of a variable
• Variable – A characteristic that is observed or
manipulated – Can take on different values
Evidence-based Chiropractic 3 © 2006
Statistical terms (cont.)
• 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
Evidence-based Chiropractic 4 © 2006
Statistical terms (cont.)
• Parameters– Summary data from a population
• Statistics– Summary data from a sample
Evidence-based Chiropractic 5 © 2006
Populations
• A population is the group from which a sample is drawn – e.g., headache patients in a chiropractic
office; automobile crash victims in an emergency room
• In research, it is not practical to include all members of a population
• Thus, a sample (a subset of a population) is taken
Evidence-based Chiropractic 6 © 2006
Random samples
• Subjects are selected from a population so that each individual has an equal chance of being selected
• Random samples are representative of the source population
• Non-random samples are not representative– May be biased regarding age, severity of the
condition, socioeconomic status etc.
Evidence-based Chiropractic 7 © 2006
Random samples (cont.)
• Random samples are rarely utilized in health care research
• Instead, patients are randomly assigned to treatment and control groups – Each person has an equal chance of being
assigned to either of the groups
• Random assignment is also known as randomization
Evidence-based Chiropractic 8 © 2006
Descriptive statistics (DSs)
• 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
Evidence-based Chiropractic 9 © 2006
DSs (cont.)
– 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
Evidence-based Chiropractic 10 © 2006
Hypothetical study data(partial from book)
Case # Visits11 77
22 2 2 33 2 2 44 3 3 55 4 4 66 3 3 77 5 5 88 3 3 99 4 4 1010 6 6 1111 2 2 1212 3 3 1313 7 7 1414 4 4
• Distribution provides a summary of:– Frequencies of each of the values
• 2 – 3• 3 – 4 • 4 – 3 • 5 – 1• 6 – 1• 7 – 2
– Ranges of values • Lowest = 2• Highest = 7
etc.
Evidence-based Chiropractic 11 © 2006
Frequency distribution table
Frequency Percent Cumulative %• 2 3 21.4 21.4• 3 4 28.6 50.0 • 4 3 21.4 71.4• 5 1 7.1 78.5• 6 1 7.1 85.6• 7 2 14.3
100.0
Evidence-based Chiropractic 12 © 2006
Frequency distributions are often depicted by a histogram
Evidence-based Chiropractic 13 © 2006
Histograms (cont.)
• A histogram is a type of bar chart, but there are no spaces between the bars
• Histograms are used to visually depict frequency distributions of continuous data
• Bar charts are used to depict categorical information – e.g., Male–Female, Mild–Moderate–Severe,
etc.
Evidence-based Chiropractic 14 © 2006
Measures of central tendency
• 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
Evidence-based Chiropractic 15 © 2006
Formula to calculate the mean
• 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
N
X
n
XX
Evidence-based Chiropractic 16 © 2006
Measures of central tendency (cont.)
• Mode – The most frequently
occurring value in a series
– The modal value is the highest bar in a histogram
ModeMode
Evidence-based Chiropractic 17 © 2006
Measures of central tendency (cont.)
• 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
Evidence-based Chiropractic 18 © 2006
Measures of central tendency (cont.)
• 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
Evidence-based Chiropractic 19 © 2006
Levels of measurement
• 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 – Examples
• Gender (e.g., male, female) • Treatment preference (e.g., manipulation,
mobilization, massage)
Evidence-based Chiropractic 20 © 2006
Levels of measurement (cont.)
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)
Evidence-based Chiropractic 21 © 2006
Levels of measurement (cont.)
• 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
Evidence-based Chiropractic 22 © 2006
Levels of measurement (cont.)
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
Evidence-based Chiropractic 23 © 2006
Levels of measurement (cont.)
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
Evidence-based Chiropractic 24 © 2006
Levels of measurement (cont.)
• 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
Evidence-based Chiropractic 25 © 2006
Levels of measurement (cont.)
• NOIR is a mnemonic to help remember the names and order of the levels of measurement– Nominal
OrdinalIntervalRatio
Evidence-based Chiropractic 26 © 2006
Levels of measurement (cont.)
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
Evidence-based Chiropractic 27 © 2006
The shape of data
• 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)
Evidence-based Chiropractic 28 © 2006
The shape of data (cont.)
Line depicting the shape of the data
Line depicting the shape of the data
Evidence-based Chiropractic 29 © 2006
The normal distribution
• 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
Evidence-based Chiropractic 30 © 2006
The normal distribution (cont.)
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
Evidence-based Chiropractic 31 © 2006
The normal distribution (cont.)
Mean = Median = ModeMean = Median = Mode
Evidence-based Chiropractic 32 © 2006
Skewed distributions
• 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
Evidence-based Chiropractic 33 © 2006
Skewed distributions (cont.)
• Skew is always toward the direction of the longer tail– Positive if skewed to the right– Negative if to the left
The mean is shifted the most
Evidence-based Chiropractic 34 © 2006
Skewed distributions (cont.)
• 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
Evidence-based Chiropractic 35 © 2006
More properties of normal curves
• 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
Evidence-based Chiropractic 36 © 2006
More properties of normal curves (cont.)
Evidence-based Chiropractic 37 © 2006
Standard deviation (SD)
• SD is a measure of the variability of a set of data
• 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
• Variance (S2) is another measure of spread
Evidence-based Chiropractic 38 © 2006
SD (cont.)
• In effect, 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
Evidence-based Chiropractic 39 © 2006
SD (cont.)
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
Evidence-based Chiropractic 40 © 2006
Distances ages deviate above and below the mean
Adding deviations always equals zero
Adding deviations always equals zero
Etc.
Evidence-based Chiropractic 41 © 2006
Calculating S2
• 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
Evidence-based Chiropractic 42 © 2006
Calculating S2 cont.
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
Evidence-based Chiropractic 43 © 2006
Calculating SD with Excel
Enter values in a column Enter values in a column
Evidence-based Chiropractic 44 © 2006
SD with Excel (cont.)
Click Data Analysis on the Tools menu
Click Data Analysis on the Tools menu
Evidence-based Chiropractic 45 © 2006
SD with Excel (cont.)
Select Descriptive Statistics and click OK
Select Descriptive Statistics and click OK
Evidence-based Chiropractic 46 © 2006
SD with Excel (cont.)
Click Input Range icon Click Input Range icon
Evidence-based Chiropractic 47 © 2006
SD with Excel (cont.)
Highlight all the values in the column
Highlight all the values in the column
Evidence-based Chiropractic 48 © 2006
SD with Excel (cont.)
Check if labels are in the first row
Check if labels are in the first row
Check Summary Statistics
Check Summary Statistics
Click OKClick OK
Evidence-based Chiropractic 49 © 2006
SD with Excel (cont.)
SD is calculated preciselyPlus several other DSs
SD is calculated preciselyPlus several other DSs
Evidence-based Chiropractic 50 © 2006
Wide spread results in higher SDs narrow spread in lower SDs
Evidence-based Chiropractic 51 © 2006
Spread is important when comparing 2 or more group means
It is more difficult to see a clear distinctionbetween groups in the upper example because the spread is wider, even though the means are the same
Evidence-based Chiropractic 52 © 2006
z-scores
• 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
X
z
Evidence-based Chiropractic 53 © 2006
z-scores (cont.)
• 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
Evidence-based Chiropractic 54 © 2006
z-scores (cont.)
Refer to a z-tableto find proportionunder the curve
Refer to a z-tableto find proportionunder the curve
Evidence-based Chiropractic 55 © 2006
z-scores (cont.)
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