Introduction to Data Display

Introduction to data display

Useful questions to ask when considering how to display information • What do you want to show? • What methods are available for this? • Is the method chosen the best? Would another have been better? Recommendations for the presentation of numbers • When summarizing categorical data, both frequencies and percentages can be used. However, if percentages are reported, it is important that the denominator (i.e. total number of observations) is given. • To summarize continuous numerical data, one should use the mean and standard deviation, or if the data have a skewed distribution use the median and range or interquartile range. Recommendations when presenting data and results in tables • Tables, including column and row headings, should be clearly labeled and a brief summary of the contents of a table should always be given in words, either as part of the title or in the main body of the text. • The amount of information should be maximized for the minimum amount of ink.

Recommendations for construction of graphs • The amount of information should be maximized for the minimum amount of ink. • Each graph should have a title explaining what is being displayed. • Axes should be clearly labeled. • Gridlines should be kept to a minimum. • Avoid three-dimensional graphs as these can be difficult to read. • The number of observations should be included.

Table or graph? Examples for badly displayed data

Describing categorical data

Clustered data

Displaying quantitative data Tables for multiple outcome measures Stem and leaf plot Histogram Showing distribution Skewed data Box–whisker plots Displaying the relationship between two continuous variables

Regression

ROC curve Tabulating categorical outcomes Tabulating the results of logistic regression analysis Tabulating quantitative outcomes Tabulating the results of regression analyses Patient flow diagram Forest plots Funnel plots Survival

Displaying results in presentations

• Keep slides simple. • Text is meant to be read. Ensure that your slides are legible. • For slides use light text on a dark background. • Keep information layout, colors, patterns, text styles, and transitions and build effects consistent for all slides in a presentation. • Maximum of six lines per slide and six words per line. • Use graphics and animation effects sparingly. • San serif fonts such as Arial are the more legible for slides. • Use a minimum font size of 28 points for titles and 18 points for the body of text

Thanks for your attention Statistical Methods Descriptive vs. Inferential

• Descriptive statistics summarize your group. – average age 78.5, 89.3% white. • Inferential statistics use the theory of probability to make inferences about larger populations from your sample. – White patients were significantly older than black and Hispanic patients, P<0.001.

48 Enter your data with statistical analysis in mind. • For small projects enter data into Microsoft Excel or directly into SPSS. • For large projects, create a database with Microsoft Access. • Keep variables names in the first row, with <=8 characters, and no internal spaces. • Enter as little text as possible and use codes for categories, such as 1=male, 2=female.

49 Screen your data thoroughly for errors and inconsistencies before doing ANY analyses. • Check the lowest and highest value for each variable. – For example, age 1-777. • Look at histograms to detect typos. • Cross-check variables to detect impossible combinations. – For example, pregnant males, survivors discharged to the morgue, patients in the ICU for 25 days with no complications.

50 Analyze, descriptive statistics, frequencies, select the variable

AGE 700

600

500

400

Statistics 300 AGE N Valid 933 200 Missing 0 Mean 79.292 Std. Dev = 26.54 100 Median 81.300 Mean = 79.3 Mode 90.0 N = 933.00

Frequency 0 Std. Dev iation 26.537 25.0 75.0 125.0175.0225.0275.0325.0375.0425.0475.0525.0575.0625.0675.0725.0775.0 Range 763.0 Minimum 14.0 Maximum 51 777.0 AGE Analyze, Descriptive Statistics, Crosstabs

SURVIVAL * 48-DISPOSITION Crosstabulation

Count 48-DISPOSITION AMA DISCHAR HOME GE REHABILI SKILLED WITH AGAINST TATION OTHER NURSING ASSISTA MEDICAL HOME FACILITY HOSPITAL MORGUE FACILITY NCE ADVICE 8 Total SURVIVAL EXPIRED 63 63 SURVIVED 224 56 12 201 236 3 138 870 Total 224 56 12 63 201 236 3 138 52933 Correct the data in the original database or spreadsheet and import a revised version into the statistical package.

• The age of 777 should be checked and

changed to the correct age.

• Suspicious values, such as an age of 106

should be checked. In this case it is correct.

53 Run descriptive statistics to summarize your data.

SURVIVAL

Valid Cumulativ Frequency Percent Percent e Percent Valid EXPIRED 63 6.8 6.8 6.8 SURVIVED 870 93.2 93.2 100.0 Total 933 100.0 100.0

49-DAYS IN HOSPITAL 400

Statistics

49-DAYS IN HOSPITAL 300 N Valid 933 Missing 0 Mean 23.34 Median 19.00 200 Mode 20 Std. Dev iation 18.03 Range 236 100 Minimum 1 Maximum 237 Std. Dev = 18.03 Mean = 23.3 N = 933.00

Frequency 0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0

54 49-DAYS IN HOSPITAL P Value

• A P value is an estimate of the probability of results such as yours could have occurred by chance alone if there truly was no difference or association. • P < 0.05 = 5% chance, 1 in 20. • P <0.01 = 1% chance, 1 in 100. • Alpha is the threshold. If P is < this threshold, you consider it statistically significant.

55 Univariate vs. Multivariate

• Univariate analysis usually refers to one predictor variable and one outcome variable – Is gender a predictor of pneumonia? • Multivariate analysis usually refers to more than one predictor variable or more than one outcome variable being evaluated simultaneously. – After adjusting for age, is gender a predictor of pneumonia?

56 Difference vs. Association

• Some tests are designed to assess whether there are statistically significant differences between groups. – Is there a statistically significant difference between the age of patients with and without pneumonia? • Some tests are designed to assess whether there are statistically significant associations between variables. – Is the age of the patient associated with the number of days in the hospital?

57 Unmatched vs. Matched

• Some statistical tests are designed to assess groups that are unmatched or independent. – Is the admission systolic blood pressure different between men and women? • Some statistical tests are designed to assess groups that are matched or data that are paired. – Is the systolic blood pressure different between admission and discharge?

58 Level of Measurement

• Categorical vs. continuous variables – If you take the average of a continuous variable, it has meaning. • Average age, blood pressure, days in the hospital. – If you take the average of a categorical variable, it has no meaning. • Average gender, race, smoker.

59 Level of Measurement

• Nominal - categorical – gender, race, hypertensive • Ordinal - categories that can be ranked – none, light, moderate, heavy smoker • Interval - continuous – blood pressure, age, days in the hospital

60 Examples of Normal and Skewed

35-SYSTOLIC BLOOD PRESSURE FIRST ER 44-DAYS IN ICU 160 1000

140 800 120

100 600

80 400 60

40 200 Std. Dev = 3.99 Std. Dev = 27.74 20 Mean = .9 Mean = 146.9 N = 933.00 N = 925.00 Frequency 0

Frequency 0 60.070.080.090.0100.0110.0120.0130.0140.0150.0160.0170.0180.0190.0200.0210.0220.0230.0240.0250.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 5.0 15.0 25.0 35.0 45.0 55.0 65.0

35-SYSTOLIC BLOOD PRESSURE FIRST ER 44-DAYS IN ICU

61 Commonly used statistical methods

• 1. Chi-square • 2. Logistic regression • 3. Student's t-test • 4. Fisher's exact test • 5. Kaplan-Meier method • 7. Wilcoxon rank-sum test • 8. Log-rank test • 9. Linear regression analysis

62 Commonly used statistical methods

• 10. One-way analysis of variance (ANOVA) • 12. Mann-Whitney U test • 13. Kruskal-Wallis test • 14. Repeated-measures analysis of variance • 15. Paired t-test • 16. Wilcoxon signed-rank test

63 Chi-square

• The most commonly used statistical test. • Used to test if two or more percentages are different. • For example, suppose that in a study of 933 patients with a hip fracture, 10% of the men (22/219) of the men develop pneumonia compared with 5% of the women (36/714). • What is the probability that this could happen by chance alone? • Univariate, difference, unmatched, nominal, =>2 groups, n=>20.

64 P N E U M O N IA C O M P L IC A T IO N 4 8 0 .0 0 -4 8 6 .9 9 * SE X C ro s s tabulatio n

SE X FEM MT A ota A LChi-square example L E l E 197678875PAC N B ount E SE U M N T O N IA 90.0%95.0%93.8% C% O w M ithin P L IC SE A T X IO N 480.00-486.99 223658PC R ount E SE N T 10.0%5.0%6.2% % w ithin SE X 219714933TC ota ount l 100.0%100.0%100.0% % w ithin SE X

Chi-Square Tests

Asymp.ExactExact Sig.Sig. Sig. (2-sided)(2-sided)(1-sided)Valuedf 7.197.0071Pearsonb Chi-Square 6.364.0121Continuitya Correction 6.492.0111Likelihood Ratio .010.008Fisher's Exact Test 7.189.0071Linear-by-Linear Association 933N of Valid Cases a.Computed only for a 2x2 table b. 065 cells (.0%) have expected count less than 5. The minimum expected count is 13.61. Fisher’s Exact Test

• This test can be used for 2 by 2 tables when the number of cases is too small to satisfy the assumptions of the chi-square. – Total number of cases is <20 or – The expected number of cases in any cell is <1 or – More than 25% of the cells have expected frequencies <5.

66 PNEUMONIA COMPLICATION 480.00-486.99 * CIRRHOSIS OR CHRONIC LIVER 571 Crosstabulation

CIRRHOSIS OR CHRONIC LIVER 571 PRESENTABSENTT otal 870875PNEUMONIA5ABSENTC ount 867.5875.07.5COMPLICATIONExpected Count 480.00-486.99 % within PNEUMONIA 100.0%99.4%.6%COMPLICATION 480.00-486.99 % within CIRRHOSIS OR 94.1%62.5%93.8% CHRONIC LIVER 571 55583PRESENTC ount 57.558.0.5Expected Count % within PNEUMONIA 100.0%94.8%5.2%COMPLICATION 480.00-486.99 % within CIRRHOSIS OR 37.5%5.9%6.2% CHRONIC LIVER 571 925933T8C otal ount 925.0933.08.0Expected Count % within PNEUMONIA 100.0%99.1%.9%COMPLICATION 480.00-486.99 % within CIRRHOSIS OR 100.0%100.0%100.0% CHRONIC LIVER 571 Chi-Square Tests

Asymp.ExactExact Sig.Sig. Sig. (2-sided)(2-sided)(1-sided)Valuedf 13.545.0001Pearsonb Chi-Square 8.674.0031Continuitya Correction 6.842.0091Likelihood Ratio .010.010Fisher's Exact Test 13.531.0001Linear-by-Linear Association 933N of Valid Cases a.Computed only for a 2x2 table b.167 cells (25.0%) have expected count less than 5. The minimum expected count is .50. Student’s t-test

• Used to compare the average (mean) in one group with the average in another group. • Is the average age of patients significantly different between those who developed pneumonia and those who did not? • Univariate, Difference, Unmatched, Interval, Normal, 2 groups.

68 Independent Samples Test

Levene's Test for Equality t-test forof EqualityVariances of Means 95% Confidence Interval of the Difference Std.MeanSig. Error DifferenceDifference(2-tailed)LowerUpperSig.dfFt -1.561-2.849-6.4291.9371.825.164.119.732931AGEEqual variances assumed 72.574-2.085-2.849-5.5721.366-.125.041Equal69 variances not assumed Mann-Whitney U test

• Same as the Wilcoxon rank-sum test • Used in place of the Student’s t-test when the data are skewed. • A nonparametric test that uses the rank of the value rather than the actual value. • Univariate, Difference, Unmatched, Interval, Nonnormal, 2 groups. 70 Paired t-test

• Used to compare the average for measurements made twice within the same person - before vs. after. • Used to compare a treatment group and a matched control group. • For example, Did the systolic blood pressure change significantly from the scene of the injury to admission? • Univariate, Difference, Matched, Interval, Normal, 2 groups.

71 Wilcoxon signed-rank test • Used to compare two skewed continuous variables that are paired or matched. • Nonparametric equivalent of the paired t-test. • For example, “Was the Glasgow Coma Scale score different between the scene and admission?” • Univariate, Difference, Matched, Interval, Nonnormal, 2 group.

72 ANOVA

One-way used to compare more than 3 means from independent groups. “Is the age different between White, Black, Hispanic patients?” Two-way used to compare 2 or more means by 2 or more factors. “Is the age different between Males and Females, With and Without Pnuemonia?”

73 Tests of Between-Subjects Effects

Dependent Variable: AGE Ty pe III Sum of Mean Source Squares df Square F Sig. Model 5769944a 4 1442486 8664.775 .000 SEX 1981.683 1 1981.683 11.904 .001 PNEUMON 1299.320 1 1299.320 7.805 .005 SEX * PNEUMON 519.282 1 519.282 3.119 .078 Error 154657.2 929 166.477 Total 5924601 933 a. R Squared = .974 (Adjusted R Squared = .974)

74 Kruskal-Wallis One-Way ANOVA

• Used to compare continuous variables that are not normally distributed between more than 2 groups. • Nonparametric equivalent to the one-way ANOVA. • Is the length of stay different by ethnicity? • Analyze, nonparametric tests, K independent samples.

75 Repeated-Measures ANOVA

• Used to assess the change in 2 or more continuous measurement made on the same person. Can also compare groups and adjust for covariates. • Do changes in the vital signs within the first 24 hours of a hip fracture predict which patients will develop pneumonia? • Analyze, General Linear Model, Repeated Measures.

76 Pearson Correlation

• Used to assess the linear association between two continuous variables. – r=1.0 perfect correlation – r=0.0 no correlation – r=-1.0 perfect inverse correlation

• Univariate, Association, Interval

77 Correlations

35-SYSTO NUMBER 43-TOTAL LIC OF NUMBER BLOOD 35-GLASG 49-DAYS COMORB OF PRESSU OW COMA IN IDITES COMPLIC RE FIRST SCALE 35-PULSE AGE HOSPITAL (0-9) ATIONS ER FIRST ER FIRST ER AGE Pearson Correlation 1.000 .088** .211** .137** .149** -.030 -.008 Sig. (2-tail ed) . .007 .000 .000 .000 .356 .809 N 933 933 933 933 925 926 923 49-DAYS IN HOSPITAL Pearson Correlation .088** 1.000 .167** .453** .039 .016 .022 Sig. (2-tail ed) .007 . .000 .000 .237 .633 .499 N 933 933 933 933 925 926 923 NUMBER OF Pearson Correlation .211** .167** 1.000 .222** .034 -.079* .055 COMORBIDITES (0-9) Sig. (2-tail ed) .000 .000 . .000 .296 .017 .093 N 933 933 933 933 925 926 923

43-TOTAL NUMBER Pearson Correlation .137** .453** .222** 1.000 -.033 -.028 .046 OF COMPLICATIONS Sig. (2-tail ed) .000 .000 .000 . .310 .393 .161 N 933 933 933 933 925 926 923 35-SYSTOLIC BLOOD Pearson Correlation .149** .039 .034 -.033 1.000 .043 .069* PRESSURE FIRST ER Sig. (2-tail ed) .000 .237 .296 .310 . .196 .035 N 925 925 925 925 925 925 923 35-GLASGOW COMA Pearson Correlation -.030 .016 -.079* -.028 .043 1.000 -.100** SCALE FIRST ER Sig. (2-tail ed) .356 .633 .017 .393 .196 . .002 N 926 926 926 926 925 926 923 35-PULSE FIRST ER Pearson Correlation -.008 .022 .055 .046 .069* -.100** 1.000 Sig. (2-tail ed) .809 .499 .093 .161 .035 .002 . N 923 923 923 923 923 923 923 **. Correlation is signif icant at the 0.01 lev el (2-tail ed). *. Correlation is signif icant at the 0.05 lev el (2-tail ed). 78 Spearman rank-order correlation • Use to assess the relationship between two ordinal variables or two skewed continuous variables. • Nonparametric equivalent of the Pearson correlation. • Univariate, Association, Ordinal (or skewed).

79 Correlations

35-SYSTO NUMBER 43-TOTAL LIC OF NUMBER BLOOD 35-GLASG 49-DAYS COMORB OF PRESSU OW COMA IN IDITES COMPLIC RE FIRST SCALE 35-PULSE AGE HOSPITAL (0-9) ATIONS ER FIRST ER FIRST ER Spearman's rho AGE Correlation Coef f icient 1.000 .089** .158** .145** .091** -.146** -.008 Sig. (2-tail ed) . .007 .000 .000 .005 .000 .806 N 933 933 933 933 925 926 923 49-DAYS IN HOSPITAL Correlation Coef f icient .089** 1.000 .142** .389** .073* .048 .037 Sig. (2-tail ed) .007 . .000 .000 .027 .149 .268 N 933 933 933 933 925 926 923 NUMBER OF Correlation Coef f icient .158** .142** 1.000 .229** .037 -.091** .042 COMORBIDITES (0-9) Sig. (2-tail ed) .000 .000 . .000 .257 .006 .202 N 933 933 933 933 925 926 923

43-TOTAL NUMBER Correlation Coef f icient .145** .389** .229** 1.000 -.014 -.076* .043 OF COMPLICATIONS Sig. (2-tail ed) .000 .000 .000 . .676 .020 .196 N 933 933 933 933 925 926 923 35-SYSTOLIC BLOOD Correlation Coef f icient .091** .073* .037 -.014 1.000 .079* .080* PRESSURE FIRST ER Sig. (2-tail ed) .005 .027 .257 .676 . .017 .015 N 925 925 925 925 925 925 923 35-GLASGOW COMA Correlation Coef f icient -.146** .048 -.091** -.076* .079* 1.000 -.038 SCALE FIRST ER Sig. (2-tail ed) .000 .149 .006 .020 .017 . .252 N 926 926 926 926 925 926 923 35-PULSE FIRST ER Correlation Coef f icient -.008 .037 .042 .043 .080* -.038 1.000 Sig. (2-tail ed) .806 .268 .202 .196 .015 .252 . N 923 923 923 923 923 923 923 **. Correlation is signif icant at the .01 lev el (2-tail ed). *. Correlation is signif icant at the .05 lev el (2-tail ed).

80 Summary of Inferential Tests

81 Unpaired vs. Paired

• Student’s t-test • Paired t-test • Chi-square • McNemar’s test • One-way ANOVA • Repeated-measures • Mann-Whitney U test • Wilcoxon signed-rank • Kruskal-Wallis H test • Friedman ANOVA

82 Parametric vs. Nonparametric

• Student’s t-test • Mann-Whitney U test • One-way ANOVA • Kruskal-Wallis test • Paired t-test • Wilcoxon signed-rank • Pearson correlation • Spearman’s r • Correlated F ratio • Friedman ANOVA (repeatedmeasures ANOVA)

83 A Good Rule to Follow

• Always check your results with a nonparametric. • If you test your null hypothesis with a Student’s t-test, also check it with a Mann- Whitney U test. • It will only take an extra 25 seconds.

84 Linear Regression

• Used to assess how one or more predictor variables can be used to predict a continuous outcome variable. • “Do age, number of comorbidities, or admission vital signs predict the length of stay in the hospital after a hip fracture?” • Multivariate, Association, Interval/Ordinal dependent variable.

85 Coefficientsa

Standardi zed Unstandardized Coeff icien Coeff icients ts Model B Std. Error Beta t Sig. 1 (Constant) -4.451 18.889 -.236 .814 AGE 7.136E-02 .045 .053 1.571 .117 NUMBER OF 2.606 .548 .159 4.757 .000 COMORBIDITES (0-9) 35-SYSTOLIC BLOOD 1.562E-02 .022 .024 .726 .468 PRESSURE FIRST ER 35-GLASGOW COMA 1.067 1.170 .030 .912 .362 SCALE FIRST ER 35-PULSE FIRST ER 2.581E-02 .047 .019 .554 .580 35-RESPIRATION -8.00E-02 .188 -.014 -.425 .671 RATE FIRST ER 86 a. Dependent Variable: 49-DAYS IN HOSPITAL Logistic Regression

• Used to assess the predictive value of one or more variables on an outcome that is a yes/no question.

• “Do age, gender, and comorbidities predict which hip fracture patients will develop pneumonia?”

• Multivariate, Difference, Nominal dependent variable, not time-dependent, 2 groups.

87 1 Total number of comorbidities 2 Cirrhosis

3 COPD

4 Gender

5 Age 88 Draw Conclusions

• We reject the null hypothesis. • Patients who are at high risk of developing pneumonia during their hospitalization for a hip fracture can be identified by: – total number of pre-existing conditions – cirrhosis – COPD – male gender

89 Survival Analysis

• Kaplan-Meier method – Used to plot cumulative survival • Log-rank test – Used to compare survival curves • Cox proportional-hazards – Used to adjust for covariates in survival analysis

90 Thanks for your attention Introduction to Statistics

Descriptive Analysis Review of Descriptive Stats. • Descriptive Statistics are used to present quantitative descriptions in a manageable form. • This method works by reducing lots of data into a simpler summary. Univariate Analysis • This is the examination across cases of one variable at a time. • Frequency distributions are used to group data. • One may set up margins that allow us to group cases into categories. • Examples include: – age categories – price categories – temperature categories. Distributions

Two ways to describe a univariate distribution • a table • a graph (histogram, bar chart) Distributions (con’t)

Sex No % Men 12 60

Women 8 40 Ditribution of participants of the research methodology workshop by sex total 20 100 70%

60%

50%

40%

30%

20%

10%

0% Men Women Distributions (con’t)

Workshop participants by specialty

Others Workshop participants by specialty

Nursing Microbiology Env ironmental sciences Fishery Fishery

Nursing Environmental sciences Other

Microbiology

0% 5% 10% 15% 20% 25% 30% 35% 40% Distributions (cont.)

A Frequency Distribution Table

Category Percent Under 35 9 36-45 21 46-55 45 56-65 19 66+ 6 Distributions (cont.)

A Histogram

50

40

30

20 Percent

10

0

66+

35

36-45 46-55 56-65

Under Central Tendency • An estimate of the “center” of a distribution • Three different types of estimates: – Mean – Median – Mode Mean • The most commonly used method of describing central tendency. • One basically totals all the results and then divides by the number of units or “n” of the sample. • Example: The pretest mean was determined by the sum of all the scores divided by the number of students taking the exam. Working Example (mean) • Lets take the set of scores: 11,10,8,9,12,11,6,13 • The Mean would be 80/8=10 Median • The median is the score found at the exact middle of the set. • One must list all scores in numerical order, and then locate the score in the center of the sample. • Example: if there are 500 scores in the list, score #250 would be the median. • This is useful in weeding out outliers. Working Example (median) • Lets take the set of scores: 11,10,8,9,12,11,6,13 • First line up the scores. 6, 8, 9, 10, 11, 11, 12, 13 • The middle score falls at 10.5. There are 8 scores and score #4 and #5 represent the halfway point. Mode • The mode is the most repeated score in the set of results. • Lets take the set of scores: 11,10,8,9,12,11,6,13 • Again we first line up the scores 6, 8, 9, 10, 11, 11, 12, 13 #11 is the most repeated score and is therefore labeled the mode. Dispersion • Three estimates: – Range – Mean Absolute Deviation – Standard Deviation • Standard Deviation is more accurate/detailed, because an outlier can greatly extend the range Range • The range is used to identify the highest and lowest scores. • Lets take the set of scores: 6, 8, 9, 10, 11, 11, 12, 13 • The range would be 6-13. This identifies the fact that 7 points separates the highest to the lowest score. Standard Deviation • The Standard Deviation is a value that shows the relation that individual scores have to the mean of the sample. • If scores are said to be standardized to a normal curve then there are several statistical manipulations that can be performed to analyze the data set. Standard Dev. (con’t) • Assumptions may be made about the percentage of scores as they deviate from the mean. • If scores are normally distributed, then one can assume that approximately 68% of the scores in the sample fall within one standard deviation of the mean. Approximately 95% of the scores would then fall within two standard deviations of the mean. Working Example (stand. dev.) • Lets take the set of scores: 11,10,8,9,12,11,6,13 • The mean of this sample was found to be 10. • Again we use the scores 11,10,8,9,12,11,6,13. • 11-10=1, 10-10=0, 8-10=-2, 9-10=-1,12-10=2, 11-10=1, 6-10=-4,13-10=3 Working Ex. (Stan. dev. con’t) • Square these values. 1, 0, 4, 1, 4, 1, 16, 9 • Total these values 36. • Divide 36 by 7: 5.15 • Take the square root of 5.15: 2.27 • 2.27 is your Standard Deviation. Interquartile range

• The median is the same as the 50th percentile. • The 25th and 75th percentiles are called the lower and upper quartiles. Interquartile range Definition: A set of n measurements on the variable x has been arranged in order of magnitude.

• The lower quartile (first quartile), Q1, is the value of x that exceeds one-fourth of the measurements and is less than the remaining 3/4. • The second quartile is the median.

• The upper quartile (third quartile), Q3, is the value of x that exceeds three-fourths of the measurements and is less than one-fourth. Thanks