Descriptive Statistics Calculations and Practical Application

Part 2

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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

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http://davidmlane.com/hyperstat/z_table.html

Visual of Normal Curve Z Distribution

Below plus Above = 100%

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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

S

x

S

XXZ

z score calculation

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Z Score to Percent Probabilityhttp://www.measuringusability.com/pcalcz.ph

p

% Below % Above % Tails % Within

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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 rankfor calculated Z score

ID Knee ROM Mean x x2 Z % Below % Above % Tails negative and positiveParticipant 1 84 86.80 -2.80 7.84 -1.897 2.89% 97.11% 5.78%Participant 2 85 86.80 -1.80 3.24 -1.220 11.13% 88.87% 22.26%Participant 3 89 86.80 2.20 4.84 1.491 93.20% 6.80% 13.60%Participant 4 86 86.80 -0.80 0.64 -0.542 29.39% 70.61% 58.77%Participant 5 87 86.80 0.20 0.04 0.136 55.39% 44.61% 89.22%Participant 6 87 86.80 0.20 0.04 0.136 55.39% 44.61% 89.22%Participant 7 87 86.80 0.20 0.04 0.136 55.39% 44.61% 89.22%Participant 8 88 86.80 1.20 1.44 0.813 79.19% 20.81% 41.61%Participant 9 88 86.80 1.20 1.44 0.813 79.19% 20.81% 41.61%Participant 10 87 86.80 0.20 0.04 0.136 55.39% 44.61% 89.22%

Sum 868 0.00 19.60 0.00n 10.00Mean 86.80 Max 89.00Median 87.00 Min 84.00Mode 87.00 Range 5.00

Variance Sample 2.178Std. Dev Sample 1.476Standard Error Mean 0.467

95% CI Z85.885 87.715lower upper

95% CI t85.744 87.856lower upper

S2X X

n

x

n

( )2 2

1 1

SX X

n

x

n

( )2 2

1 1SE SS

nm X or

ZX X

S

x

S

value ZTabledX

n

S

t valueTabledX

n

S

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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

SE SS

nm X or

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4or Xm SSE

33.19

4or Xm SSE

116

4or Xm SSE

80.25

4or Xm SSE

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4or Xm SSE

5.14

3or Xm SSE

14

2or Xm SSE

5.4

1or Xm SSE

The larger the n the smaller the SEM. The smaller the Std Dev, the smaller the SEM

Confidence Interval of the Mean for Statistical Inference About PopulationUsing 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 scoreCI 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

Z and t distributions at 95%; Statistical Inference

Z

5%

n-1

2.5% + 2.5% = 5%

t distribution

t value approximates Z when sample size

is large Degrees of freedom (df) for single group = n-1)

http://statpages.org/pdfs.html

t = Z

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

value ZTabledXn

S

95% Confidence Interval for the Mean using the Z distribution

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http://www.mccallum-layton.co.uk/stats/ConfidenceIntervalCalc.aspx

FISH with 95% Confidence interval with t distribution

t valueTabledXn

S

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

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Positive skewed not normal; the median or modemay better represent this group

Not skewed normal; the mean would represent thisgroup

Negative skewed not normal; the median or mode may better represent this group

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Positive

Negative

Stem plot

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Fit of Normal DistributionMean 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 pullednegatively towards the outliers to the left. In this case the median in red betterrepresents the density of the distribution.

Mean is not a good representation of scores because mean in green is pulledpositively towards the outliers to the left. In this case the median in red betterrepresents 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

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Positive Skewness

Negative Skewness

Too high and skinnynot normal; positive value

Too short and widenot normal; negative value

Kurtosis

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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 supportedif 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.

 

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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.

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Normal Distribution CalculatorCalculated using, mean, median, SD, range, skewness and kurtosis

Excel Descriptive Statistics

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Variable NameKnee ROM

Mean 86.80Standard Error 0.47

Median 87.00Mode 87.00

Standard Deviation 1.48Sample Variance 2.18

Kurtosis 0.26Skewness -0.61

Range 5.00Minimum 84.00Maximum 89.00

Sum 868.00Count 10

Tabled t value 2.26Confidence Level (95.0%) 1.06

Upper Hinge 75% 88Lower Hinge 25% 86

Interquartile Range 2

t valueTabled

n

S

1

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

Z 95% Confidence Interval

Mean and CI

Ou

tco

me

1

86.00

86.20

86.40

86.60

86.80

87.00

87.20

87.40

87.60

87.80

88.00

t 95% Confidence Interval

Mean and CI

Ou

tco

me

Input Type in tan cells belowSample Mean m 86.8Standard Deviation S 1.48Sample Size n 10Significance Level two tail a 0.05

Output Confidence IntervalStandard Error of the Mean (SEM) StdErrMean 0.47tabled Z score two tail Table Z 1.96Confidence Interval two tail for Z value 85.88 87.72Degrees of Freedom df 9tabled t-value two tail table t 2.26Confidence Interval two tail for t value 85.74 87.86 SEM * Z SEM * t

0.92 1.06

Confidence Interval for the Mean

value ZTabledX

n

S

t valueTabledX

n

S

Calculator for SEM, tabled value and CI

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

X Data Please hand calculate all cells highlighted in orange

ID Pain NPS Mean x x2 Z % Below % Above % Tails negative and positiveParticipant 1 3 5.70 50.00% 50.00% 100.00%Participant 2 4 5.70 50.00% 50.00% 100.00%Participant 3 4 5.70 50.00% 50.00% 100.00%Participant 4 5 5.70 50.00% 50.00% 100.00%Participant 5 5 5.70 50.00% 50.00% 100.00%Participant 6 5 5.70 50.00% 50.00% 100.00%Participant 7 7 5.70 50.00% 50.00% 100.00%Participant 8 7 5.70 50.00% 50.00% 100.00%Participant 9 8 5.70 50.00% 50.00% 100.00%Participant 10 9 5.70 50.00% 50.00% 100.00%

Sum 57n 10Mean 5.7 MaxMedian MinMode Range

Variance Sample

Std. Dev Sample

Standard Error Mean 95% CI Z

lower upper

Raw Score Transformed to Z ScoreDependent Variable:

Raw Score X Type in score

Scale: 95% CI t Mean Score Mean Type in mean

Standard Deviation Std Dev Type in Std Dev

lower upper Z Score CalculatedZ Score Calculated Z #DIV/0!Percentile Rank Answer #DIV/0!

Input Raw Score to Create Z Score

S2X X

n

x

n

( )2 2

1 1

SX X

n

x

n

( )2 2

1 1

SE SS

nm X or

value ZTabledX

n

S

ZX X

S

x

S

t valueTabledX

n

S

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Use FISH on Part B Excel to Check Assignment Calculations

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