1. An introduction to an expansive and complex field
Biostatistics Basics
An introduction to an expansive and complex field

2. Common statistical terms
Data
Measurements or observations of a variable
Variable
A characteristic that is observed or manipulated
Can take on different values
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3. 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
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4. Statistical terms (cont.)
Parameters
Summary data from a population
Statistics
Summary data from a sample
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5. Evidence-based Chiropractic
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
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6. Evidence-based Chiropractic
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.
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7. Evidence-based Chiropractic
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
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8. 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
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9. Evidence-based Chiropractic
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
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10. Hypothetical study data (partial from book)
Case # Visits
1 7
2 2
3 2
4 3
5 4
6 3
7 5
8 3
9 4
10 6
11 2
12 3
13 7
14 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.
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11. Frequency distribution table
Frequency Percent Cumulative %
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12. Frequency distributions are often depicted by a histogram
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13. Evidence-based Chiropractic
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.
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14. 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
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15. 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
ΣX 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
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16. Measures of central tendency (cont.)
Mode
The most frequently occurring value in a series
The modal value is the highest bar in a histogram
Mode
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17. 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
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18. 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
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19. Evidence-based Chiropractic
Levels of measurement
There are 4 levels of measurement
Nominal, ordinal, interval, and ratio
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)
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20. Levels of measurement (cont.)
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)
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21. 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
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22. Levels of measurement (cont.)
Interval
Measurements are ordered (like ordinal data)
Have equal intervals
Does not have a true zero
Example
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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23. Levels of measurement (cont.)
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
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24. 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
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25. Levels of measurement (cont.)
NOIR is a mnemonic to help remember the names and order of the levels of measurement
Nominal Ordinal Interval Ratio
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26. Levels of measurement (cont.)
Measurement scale
Permissible mathematic operations
Best measure of
central tendency
Nominal
Counting
Mode
Ordinal
Greater or less than operations
Median
Interval
Addition and subtraction
Symmetrical – Mean
Skewed – Median
Ratio
Addition, subtraction,
multiplication and division
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27. Evidence-based Chiropractic
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)
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28. The shape of data (cont.)
Line depicting the shape of the data
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29. 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
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30. The normal distribution (cont.)
Mean
The highest point of the overlying normal curve is at the mean
As one moves away from
the mean in either direction
the height of the curve
decreases, approaching,
but never reaching zero
A normal distribution is symmetric about its mean
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31. The normal distribution (cont.)
Mean = Median = Mode
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32. Evidence-based Chiropractic
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
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33. 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
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34. 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
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35. 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
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36. More properties of normal curves (cont.)
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37. 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
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38. Evidence-based Chiropractic
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
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39. Evidence-based Chiropractic
SD (cont.)
Ages are spread out along an X axis
The amount ages are spread out is known as dispersion or spread
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40. Distances ages deviate above and below the mean
Etc.
Adding deviations always equals zero
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41. Evidence-based Chiropractic
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
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42. Evidence-based Chiropractic
Calculating S2 cont.
S2 is not in the same units (age), but SD is
Symbol for SD of a sample  for a population
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43. Calculating SD with Excel
Enter values in a column
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44. SD with Excel (cont.) Click Data Analysis on the Tools menu
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45. SD with Excel (cont.) Select Descriptive Statistics and click OK
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46. Evidence-based Chiropractic
SD with Excel (cont.)
Click Input Range icon
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47. SD with Excel (cont.) Highlight all the values in the column
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48. SD with Excel (cont.) Click OK Check if labels are in the first row
Check Summary Statistics
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49. SD with Excel (cont.) SD is calculated precisely
Plus several other DSs
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50. Wide spread results in higher SDs narrow spread in lower SDs
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51. Spread is important when comparing 2 or more group means
It is more difficult to see a clear distinction between groups in the upper example because the spread is wider, even though the means are the same
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52. Evidence-based Chiropractic
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
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53. Evidence-based Chiropractic
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
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54. Evidence-based Chiropractic
z-scores (cont.)
Refer to a z-table to find proportion under the curve
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55. Evidence-based Chiropractic
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
0.9441
z-scores (cont.)
Corresponds to the area
under the curve in black
0.9332
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