basic concepts chapter 1 college of education and health professions
TRANSCRIPT
Basic Concepts
Chapter 1
College of Educationand Health Professions
Statistical Inference• A Population is a group with a common
characteristic.• A population is usually large and it is difficult to
measure all members.• To make inference about a population we take a
representative sample (RANDOM).• In a random sample each member of the
population is equally likely to be selected.• A sample cannot accurately represent the
population unless it is drawn without BIAS.• In a bias free sample selection of one member does
not affect to selection of future subjects.
Types of Variables
• Continuous variable – can assume any value (ht, wt)
• Discrete variable – limited to certain values: integers or whole numbers (2.5 children?)
Level of Measurement
• Nominal Scale: mutually exclusive (male, female)• Ordinal Scale: gives quantitative order to the
variable, but it DOES NOT indicate how much better one score is than another (pain of 2 is not twice of 1)
• Interval Scale: has equal units and zero is not an absence of the variable (temperature)
• Ratio Scale: based on order, has equal distance between scale points, and zero is an absence of value
Independent & Dependent Variables
• Independent Variable: the variable being controlled (gender, group, class).
• Dependent Variable: NOT free to vary (math score, height, weight).
• The INDEPENDENT VARIABLE is controlled by the researcher – Effects of exercise on body fat.– Effects of type of instruction on learning.
• The DEPENDENT VARIABLE is the variable being studied – Body fat– learning
Parameters and Statistics
• A parameter represents the population μ.• A statistic represents the sample .• The difference between a statistic and a
parameter is the result of sampling error.
Describing and Exploring Data
Chapter 2
College of Educationand Health Professions
A frequency distribution organizes the data in a logical order.
Most of the scores are between 46 – 67.
Histogram with the normal curve superimposed.
Measures of Central Tendency
• Mean, Median and Mode• Describe the middle or central characteristics
of the data.• The mode is the most frequent score.• The median is the middle score.• In a normal distribution the mean, median and
mode are the near the same score.
Measures of Variability
• Range (lowest-highest). Suffers reliance on extreme scores.
• Interquartile Range: middle 50% of the scores.• Variance: sum of squared deviations from the
mean divided by (N-1), [data in squared units]• Standard Deviation: square root of the
variance [data in original units].
Formulas for Mean, Variance and SD
𝑋=∑ 𝑋𝑁
𝑠2=∑ (𝑋 𝑖−𝑋 )2
𝑁−1 𝑠=√∑ (𝑋 𝑖−𝑋 )2
𝑁−1
Mean Variance StandardDeviation
Degrees of Freedom
• Df is the number of things that are free to vary.
• For Example– The sum of three numbers is 10. The df is two, you
can pick any two numbers. Once you pick the first two numbers the third number is fixed, it is not free to vary.
5 + 4 + ? = 10
SPSS Box Plots
• The top of the box is the upper fourth or 75th percentile. • The bottom of the box is the lower fourth or 25th percentile. • 50 % of the scores fall within the box or interquartile range. • The horizontal line is the median. • The ends of the whiskers represent the largest and smallest values
that are not outliers. • An outlier, O, is defined as a value that is smaller (or larger) than
1.5 box-lengths.• An extreme value, E, is defined as a value that is smaller (or larger)
than 3 box-lengths.• Normally distributed scores typically have whiskers that are about
the same length and the box is typically smaller than the whiskers.
A normal distribution is symmetrical, uni-modal and mesokurtic.
Playturtic is flat.
Leptokurtic is peaked.
Tests for Normality
Tests for Normality
Tests for Normality
Tests for Normality
Tests of Normality
.175 14 .200* .890 14 .081set1Statistic df Sig. Statistic df Sig.
Kolmogorov-Smirnova
Shapiro-Wilk
This is a lower bound of the true significance.*.
Lilliefors Significance Correctiona.
– Not less than 0.05 so the data are normal.
Tests for Normality: Normal Probability Plot or Q-Q Plot
5 10 15 20 25 30
Observed Value
-2
-1
0
1
2
Exp
ecte
d N
orm
al
Normal Q-Q Plot of set1– If the data are
normal the points cluster around a straight line
Tests for Normality: Boxplots
set1
9.00
12.00
15.00
18.00
21.00
24.00
27.00
– Bar is the median, box extends from 25 – 75th percentile, whiskers extend to largest and smallest values within 1.5 box lengths
set1
0.00
20.00
40.00
60.00
80.00
100.00
16
15
– Outliers are labeled with O, Extreme values are labeled with a star
Tests for Normality: Normal Probability Plot or Q-Q Plot
5 10 15 20 25 30
Observed Value
-2
-1
0
1
2
Exp
ecte
d N
orm
al
Normal Q-Q Plot of set1
Outliers and Extreme Scores
Chapter 2
College of Educationand Health Professions
SPSS – Explore BoxPlot• The top of the box is the upper
fourth or 75th percentile. • The bottom of the box is the lower
fourth or 25th percentile. • 50 % of the scores fall within the box
or interquartile range. • The horizontal line is the median. • The ends of the whiskers represent
the largest and smallest values that are not outliers.
• An outlier, O, is defined as a value that is smaller (or larger) than 1.5 box-lengths.
• An extreme value, E , is defined as a value that is smaller (or larger) than 3 box-lengths.
• Normally distributed scores typically have whiskers that are about the same length and the box is typically smaller than the whiskers.
Choosing a Z Score to Define Outliers
Z Score % Above % +/- Above
3.0 0.0013 0.0026
3.1 0.0010 0.0020
3.2 0.0007 0.0014
3.3 0.0005 0.0010
3.4 0.0003 0.0006
Decisions for Extremes and Outliers1. Check your data to verify all numbers are entered correctly.2. Verify your devices (data testing machines) are working
within manufacturer specifications.3. Use Non-parametric statistics, they don’t require a normal
distribution.4. Develop a criteria to use to label outliers and remove them
from the data set. You must report these in your methods section.
1. If you remove outliers consider including a statistical analysis of the results with and without the outlier(s). In other words, report both, see Stevens (1990) Detecting outliers.
5. Do a log transformation.1. If you data have negative numbers you must shift the numbers to
the positive scale (eg. add 20 to each).2. Try a natural log transformation first in SPSS use LN().3. Try a log base 10 transformation, in SPSS use LG10().
Transform - Compute
Add 10 to the data. Then log transform.
Add 10 to each data point.
Try Natural Log.
Last option, use Log10.
Add 10 to each data point, since you can not take a log of a negative number.
First try a Natural Log Transformation
If Natural Log doesn’t work try Log10 Transformation.
Outlier Criteria: 1.5 * Interquartile Range from the Median
Milner CE, Ferber R, Pollard CD, Hamill J, Davis IS. Biomechanical Factors Associated with Tibial Stress Fracture in Female Runners. Med Sci Sport Exer. 38(2):323-328, 2006.
Statistical analysis. Boxplots were used to identify outliers, defined as values >1.5 times the interquartile range away from the median. Identified outliers were removed from the data before statistical analysis of the differences between groups. A total of six data points fell outside this defined range and were removed as follows: two from the RTSF group for BALR, one from the CTRL group for ASTIF, one from each group for KSTIF, and one from the CTRL group for TIBAMI.
Outlier Criteria: 1.5 * Interquartile Range from the Median (using SPSS)
Outlier Criteria: 1.5 * Interquartile Range from the Median (using SPSS)
Descriptives
-.0785 .53292
-1.1684
1.0115
-.2540
-.3549
8.520
2.91893
-5.27
10.00
15.27
3.39
1.176 .427
3.783 .833
Mean
Lower Bound
Upper Bound
95% ConfidenceInterval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
JumpStatistic Std. Error
Outliers = 1.5 * 3.39 above and below -.3549
Outlier Criteria: ± 3 standard deviations from the mean
Tremblay MS, Barnes JD, Copeland JL, Esliger DW. Conquering Childhood Inactivity: Is the Answer in the Past? Med Sci Sport Exer. 37(7):1187-1194, 2005.
Data analyses. The normality of the data was assessed by calculating skewness and kurtosis statistics. The data were considered within the limits of a normal distribution if the dividend of the skewness and kurtosis statistics and their respective standard errors did not exceed ± 2.0. If the data for a given variable were not normally distributed, one of two steps was taken: either a log transformation (base 10) was performed or the outliers were identified (± 3 standard deviations from the mean) and removed. Log transformations were performed for push-ups and minutes of vigorous physical activity per day. Outliers were removed from the data for the following variables: sitting height, body mass index (BMI), handgrip strength, and activity counts per minute..
Computing and Saving Z Scores
Check this box and SPSS creates and saves the z scores for all selected variables. The z scores in this case will be named zLight1…
Computing and Saving Z Scores
Now, you can identify and remove raw scores above and below 3 sds if you want to remove outliers.
Comparison of Outlier MethodsMedian ± 1.5 * Interquartile Range
27 ± 1.5 * 7 gives a range of 16.5 – 37.5
TABLE 1.1 Newcomb's measurements of the passage time of light
28 22 36 26 28 28
26 24 32 30 27 24
33 21 36 32 31 25
24 25 28 36 27 32
34 30 25 26 26 25
-44 23 21 30 33 29
27 29 28 22 26 27
16 31 29 36 32 28
40 19 37 23 32 29
-2 24 25 27 24 16
29 20 28 27 39 23
Comparison of Outlier Methods ± 2 SDs (Notice the 16’s are not removed)
TABLE 1.1 Newcomb's measurements of the passage time of light
28 22 36 26 28 28
26 24 32 30 27 24
33 21 36 32 31 25
24 25 28 36 27 32
34 30 25 26 26 25
-44 23 21 30 33 29
27 29 28 22 26 27
16 31 29 36 32 28
40 19 37 23 32 29
-2 24 25 27 24 16
29 20 28 27 39 23
Comparison of Outlier Methods ± 2 SDs (Notice the 16s are not removed)
TABLE 1.1 Newcomb's measurements of the passage time of light
28 22 36 26 28 28
26 24 32 30 27 24
33 21 36 32 31 25
24 25 28 36 27 32
34 30 25 26 26 25
-44 23 21 30 33 29
27 29 28 22 26 27
16 31 29 36 32 28
40 19 37 23 32 29
-2 24 25 27 24 16
29 20 28 27 39 23
Comparison of Outlier Methods ± 3 SDs (Step 1, then run Z scores again)
TABLE 1.1 Newcomb's measurements of the passage time of light
28 22 36 26 28 28
26 24 32 30 27 24
33 21 36 32 31 25
24 25 28 36 27 32
34 30 25 26 26 25
-44 23 21 30 33 29
27 29 28 22 26 27
16 31 29 36 32 28
40 19 37 23 32 29
-2 24 25 27 24 16
29 20 28 27 39 23
Comparison of Outlier Methods ± 3 SDs (Step 2, after already removing the -44, the -2 then has a SD of -4.68 so it is removed)
TABLE 1.1 Newcomb's measurements of the passage time of light
28 22 36 26 28 28
26 24 32 30 27 24
33 21 36 32 31 25
24 25 28 36 27 32
34 30 25 26 26 25
-44 23 21 30 33 29
27 29 28 22 26 27
16 31 29 36 32 28
40 19 37 23 32 29
-2 24 25 27 24 16
29 20 28 27 39 23
Conclusions?
• The median ± 1.5 * interquartile range appears to be too liberal.
• ± 2 SDs may also be too liberal and statisticians may not approve.
• An iterative process where you remove points above and below 3 SDs and then re-check the distribution may be the most conservative and acceptable method.
• Choosing 3.1, 3.2, or 3.3 as a SD increases the protection against removing a score that is potentially valid and should be retained.
Choosing a Z Score to Define Outliers
Z Score % Above % +/- Above
3.0 0.0013 0.0026
3.1 0.0010 0.0020
3.2 0.0007 0.0014
3.3 0.0005 0.0010
3.4 0.0003 0.0006
Decisions for Extremes and Outliers1. Check your data to verify all numbers are entered correctly.2. Verify your devices (data testing machines) are working
within manufacturer specifications.3. Use Non-parametric statistics, they don’t require a normal
distribution.4. Develop a criteria to use to label outliers and remove them
from the data set. You must report these in your methods section.
1. If you remove outliers consider including a statistical analysis of the results with and without the outlier(s). In other words, report both, see Stevens (1990) Detecting outliers.
5. Do a log transformation.1. If you data have negative numbers you must shift the numbers to
the positive scale (eg. add 20 to each).2. Try a natural log transformation first in SPSS use LN().3. Try a log base 10 transformation, in SPSS use LG10().
Data Transformations and Their Uses
Data Transformation Can Correct For
Log Transformation (log(X)) Positive Skew, Unequal Variances
Square Root Transformation (sqrt(X)) Positive Skew, Unequal Variances
Reciprocal Transformation (1/X) Positive Skew, Unequal Variances
Reverse Score Transformation – all of the above can correct for negative skew, but you must first reverse the scores. Just subtract each score from the highest score in the data set + 1.
Negative Skew