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Descriptive Statistics Calculations and Practical Application Part 2 1

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Content Normal Z distribution Z Score Calculation and Application Z and t distributions 95% confidence interval of the mean Friendly Introductory Statistics Help (FISH) More Descriptive Graphics Test for Normal Distribution Hand Calculations 2

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http://davidmlane.com/hyperstat/z_table.html Visual of Normal Curve Z Distribution Below plus Above = 100% 4

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Z Score A Z score takes a raw score and converts it to a number that expresses how far that value is from the mean in standard deviation units A Z score can be positive, above the mean, negative, below the mean, and 0 equal to the mean A Z score can represent a percentile and probability value The following is a formula to calculate a Z score where X = raw score, X bar = mean and S = standard deviation Z scores explained z score calculation 5

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Z Score to Percent Probability http://www.measuringusability.com/pcalcz.php % Below% Above % Tails % Within 6

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Example Answers Example from excel spreadsheet on descriptive statistics with filled in values for calculations. Note the percentages are calculated for you. % below = percentile rank for calculated Z score 7

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What is Standard Error of the Mean: (error) standard error of the mean is an estimate of the amount that an obtained mean may be expected to differ by chance from the true population mean. http://medical- dictionary.thefreedictionary.com/standard+error+of+the+mean http://medical- dictionary.thefreedictionary.com/standard+error+of+the+mean The larger the n the smaller the SE M. The smaller the Std Dev, the smaller the SE M

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Confidence Interval of the Mean for Statistical Inference About Population Using Z score CI 95% = Mean ± 1.96 x (standard error mean) CI 95% = 86.8 ± 1.96 x (0.467) CI 95% = 86.8 ±.915 CI 95% = 85.89 to 87.2 Using t score CI 95% = Mean ± t value x (standard error mean) CI 95% = 86.8 ± 2.2621 x (0.467) CI 95% = 86.8 ± 1.06 CI 95% = 85.74 to 87.862

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Z and t distributions at 95%; Statistical Inference Z 5% n-1 2.5% + 2.5% = 5% t distribution

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t value approximates Z when sample size is large D egrees of freedom (df) for single group = n-1) http://statpages.org/pdfs.html t = Z

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Example using Friendly Introductory Statistics Help (FISH), enter data STEP 1 and perform STEPS 2-10 to check your calculations.

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95% Confidence Interval for the Mean using the Z distribution 13 http://www.mccallum-layton.co.uk/stats/ConfidenceIntervalCalc.aspxwww.mccallum-layton.co.uk/stats/ConfidenceIntervalCalc.aspx

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FISH with 95% Confidence interval with t distribution

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Normal distribution should have density in the middle central values like the distribution shown in this table 15

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Positive skewed not normal; the median or mode may better represent this group Not skewed normal; the mean would represent this group Negative skewed not normal; the median or mode may better represent this group 16 Positive Negative

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Stem plot 17

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Fit of Normal Distribution Mean is a good representation of scores because mean, median, and mode (as shown by the yellow arrow) are at center of the distribution within a distribution that is reasonably bell shaped. Mean is not a good representation of scores because mean in green is pulled negatively towards the outliers to the left. In this case the median in red better represents the density of the distribution. Mean is not a good representation of scores because mean in green is pulled positively towards the outliers to the left. In this case the median in red better represents the density of the distribution. http://bcs.whfreeman.com/ips4e/pages/bcs-main.asp?s=00010&n=99000&i=99010.01&v=category&o=&ns=0&t=&uid=0&rau=0 18 Positive Skewness Negative Skewness

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Too high and skinny not normal; positive value Too short and wide not normal; negative value Kurtosis 19

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Measures for Normality Skewness (If skewness divided by its error is greater than +1.96 or less than – 1.96 then skewness could cause data to fail normal distribution) 0 if mean and median equal Positive if mean is greater than median Negative if mean is less than median Kurtosis (If kurtosis divided by its error is greater than +1.96 or less than – 1.96 then kurtosis could cause data to fail normal distribution) Mesokurtic is normal =0 Leptokurtic is high and skinny = positive value Platykurtic is short and wide = negative value Normality test : Shapiro-Wilk test Significance level: alpha > 0.05 Inference: Null Hypothesis Retained: hypothesis that data does not differ from the theoretical normal distribution is supported if the significance level is greater than.05; data are normally distributed; it is therefore OK to use the mean as representative of a given group or time. 20

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If p value is greater than level of significance then accept that your data are normally distributed. That is, the data do not significantly differ from the normal distribution. 21 Normal Distribution Calculator Calculated using, mean, median, SD, range, skewness and kurtosis

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Excel Descriptive Statistics 22 Variable Name Knee ROM Mean86.80 Standard Error0.47 Median87.00 Mode87.00 Standard Deviation1.48 Sample Variance2.18 Kurtosis0.26 Skewness-0.61 Range5.00 Minimum84.00 Maximum89.00 Sum868.00 Count10 Tabled t value 2.26 Confidence Level (95.0%)1.06 Upper Hinge 75%88 Lower Hinge 25%86 Interquartile Range2

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Calculator for SEM, tabled value and CI

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Proceed to Excel Workbooks to develop your understanding and application of this content 24

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Use FISH on Part B Excel to Check Assignment Calculations 25

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