CTRC Core Curriculum Seminar Series

Descriptive Statistics: Data Types and Measures, Central

Tendency, Variability

Chang-Xing Ma, PhDAssociate Professor

Department of Biostatistics, UB

January 4, 2012

Disclosure Statement

• Chang-Xing Ma, PhD– Nothing to disclose

Goals and Objectives

• Goals: Gain the knowledge of basic statistics and how to describe the data

• Objectives: – Describe the data type– Summarize data – Understand Measure of Central Tendency– Understand Measure of Dispersion

Outline

• Basic concepts of biostatistics• Data type• Summarize data• Measure of Central Tendency• Measure of Dispersion

Some terminology

• Statistics is the study of how to collect, organize, analyze, and interpret numerical information from data

• Biostatistics—the theory and techniques for collecting, describing, analyzing, and interpreting health data.

Some terminology

• Population refer to all measurements or observations of interest

• Sample is simply a part of the population. But the sample MUST represent the population. – A random sample is such a representative sample

• The sample must be large enough• The sample should be selected randomly

Some terminology

• Parameter is some numerical or nominal characteristic of a population– A parameter is constant, e.g. mean of a population– Usually unknown

• Statistic is some numerical or nominal characteristic of a sample.– We use statistic as an estimate of a parameter of the

population– It tends to differ from one sample to another– We also use statistic to test hypothesis

Population: all U.S. persons ~ Normal (µh,σh2),

A random sample: sample size =

Gender Height Weight

mean height:

mean weight

Parameters

A sample

std height:

std weight

statistics

% of male (=1)

(µw,σw2),

Generate

True Parameters

Sources of data

Records Surveys Experiments

Comprehensive Sample

Quantitative

continuous

Types of variables

Quantitative variables Qualitative variables

Quantitative

discrete

Qualitative

nominal

Qualitative ordinal

Data Types

• Numerical (Quantitative)– numerical measurement

• Height• Weight

• Categorical (Qualitative)– with no natural sense of ordering

• Gender• Hair color • Blood type

Numerical Variable

• Continuous– Range of values

• Height in inch

• Discrete– Limited possible values

• # of smoking per day• # of children in a family

• Age -

• Ordinal (Categorical) vs. Discrete (Numerical)• Ordinal

– Cancer Stage I, II, III, IV– Stage II ≠ 2 times Stage I– Categories could also be A, B, C, D

• Discrete– # of children: 0, 1, 2, …– 4 children = 2 times 2 children

Determining Data Types

Descriptive Statistics – reducing a complex mass of data to a manageable set of information

• Descriptive Statistics: the summary and presentation of data to:– simplify the data– enable meaning full interpretation– support decision making

• Numerical descriptive measures (few numbers)

• Graphical presentations

Inferential statistics

From a sample • to estimate population parameters• to test hypothesis • to build the model to reflect the population• …

The student test score (FCAT)

Student ID Race Sex Reading Math PovertyCode:

Race:W – WhiteB – BlackH – HispanicA – Asian

Sex:F – FemaleM – Male

Poverty:0 – not poor1 – poor

Problem 1

1.Among the 6 variables, which ones are qualitative and which ones are quantitative?2.Is Race nominal or ordinal?

Descriptive Statistics

• Categorical variables: – Frequency distribution– Bar chart, pie chart– Contingency tables

• Continuous variables:– Grouped frequency table– Central Tendency– Variability

Simple Frequency DistributionAn ordered arrangement that shows the

frequency of each level of a variable.race Frequency Percent-----------------------------A 7 4.07 B 42 24.42 H 8 4.65 W 115 66.86

sex Frequency Percent----------------------------F 86 50.00 M 86 50.00

Simple Frequency Distribution

• It is useful for categorical variable• For continuous variable,

– it allows you to pick up at a glance some valuable information, such as highest, lowest value.

– ascertain the general shape or form of the distribution

– make an informed guess about central tendency values

Bar Chart

• summarizing a set of categorical data - nominal or ordinal data

• It displays the data using a number of rectangles, each of which represents a particular category. The length of each rectangle is proportional to the number of cases in the category it represents

• can be displayed horizontally or vertically

• they are usually drawn with a gap between the bars

• Bars for multiple (usually two) variables can be drawn together to see the relationship

0

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A B H W

Race

BY

Horizontally

Pie Chart

• summarizing a set of categorical data - nominal or ordinal data

• It is a circle which is divided into segments.

• Each segment represents a particular category.

• The area of each segment is proportional to the number of cases in that category.

Female Male

Complex frequency distribution Table

Distribution of 20 lung cancer patients at the chest department of Alexandria hospital and 40 controls in May 2008 according to smoking

Smoking

Lung cancerTotal

Cases Control

No. % No. % No. %

Smoker15 75% 8 20% 23

38.33

Non smoker

5 25% 32 80% 3761.6

7

Total 20 100 40 100 60 100

How about continuous variables?

• How data is distributed?

• Measure of Central Tendency

• Measure of Variability

Grouped Frequency Distribution – for continuous variable

DATA: Frequency Table

Interval Size:

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150 165 180 195 210 225 240 255 270 285

N:µ:σ:

New Data

HISTOGRAM

POLYGON

15

Example Data

Grouped Frequency Distribution

• BUT the problem is that so much information is presented that it is difficult to discern what the data is really like, or to "cognitively digest" the data.

• the simple frequency distribution usually need to condense even more. – It is possible to lose information (precision) about the data to gain

understanding about distributions. • This is the function of grouping data into equal-sized intervals

called class intervals.• The grouped frequency distribution is further presented as

Frequency Polygons, Histograms, Bar Charts, Pie Charts.

Describing Distributions• Bell-Shaped Distribution

– Normal distribution N (µ=0, σ2 =1)

– t-distribution

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Describing Distributions• Skewed Distribution – positively skewed distribution

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Describing Distributions• Skewed Distribution – negatively skewed distribution

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Describing Distributions• Other Shapes

Rectangular Bimodal

Describing Distributions• Other Shapes

J-curve

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Probability density function - Normal

green curve isstandard normaldistribution

z-transform

Measure of Central TendencyMean, Median, Mode

• The Mean– average value– not robust to outlying value

• Length of hospital stays:6, 4, 5, 9, 10, 7, 1, 4, 3, 4

• Mean=(6+4+5+9+10+7+1+4+3+4)/10=5.3

N

XX

N

ii

1

Measure of Central TendencyMean, Median, Mode

• The Median– is the point that divides a distribution of data into

two equal parts– robust to outlying value

• Length of hospital stays: sort data1 3 4 4 4 5 6 7 9 10

• median=4.5Split Data

Measure of Central TendencyMean, Median, Mode

• The Mode– is the midpoint of the interval that has highest

frequency– robust to outlying value, but sometimes

misleading• Length of hospital stays: sort data

1 3 4 4 4 5 6 7 9 10

• Mode=4, which occurred 3 times.Most frequently

Comparison between mean and median

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Mean Median

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Comparison between mean and median

MeanMedian

0 5 10 15 20 25 300

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Comparison between mean and median

Mean Median

Summary

• Frequency distribution• Histogram, Polygon graph• Bar Chart, Pie Chart• Describing Distributions• Mean, Median, Mode

DATASET: http://128.205.94.145/STA2008/FL_School0022.xls

Problem 2

• In a study, we collected a medical measurements X for 4 patients

• Data of X: 2, 3, 5, 6

• Mean of X? • Median of X?• Mode of ?

Descriptive StatisticsVariability

• The sample range• Interquartile range• The sample standard deviation (SD), variance• Standard error of mean (SEM)

Measures of Dispersion - Range

• Range – the difference between the lowest and highestFor example, Age of Patients (years): 6 13 7 14 10 14 15 9 7 2 7 13 16 9 8 3 3 17 8 5 4 9 9 6lowest 2, highest 17Range=2 -17 years

• When sample size increases, the range tends to increase as well. (not robust)

Measures of Dispersion - Range

• All of curves have the same range

• Mean?• Median?

Measures of DispersionPercentiles, Deciles, Quartiles

• Percentiles: based on dividing a sample or population into 100 equal parts.

• Deciles divide the distribution into 10 parts• Quartiles divide the distribution into 4 equal parts.

– 1st quartile includes the lowest 25% of the values (Q1)– 2st quartile includes the values from 26 percentile through 50

percentile (Q2) - median– 3st quartile includes the values from 51 percentile through 75

percentile (Q3)

Measures of DispersionInterquarile Range

• Interquarile Range – the 25 percentile (1st quartile) to 75 percentile (3rd quartile)

• Age of Patients (years): 2 3 3 4 5 6 6 7 7 7 8 8 9 9 9 9 10 13 13 14 14 15 16 17– 1st quartile 6, 2nd quartile 8.5, 3rd 13– Interquarile Range = 6 -13 years

• Interquarile Range is a robust estimate of data variability

Measures of DispersionInterquarile Range

Robust estimate, less efficient

Deviations from the meanVariance and Standard Deviation

• deviation: observation - mean• “sum” of deviation

)( xxi 0)( xxiBUT

Deviations from the meanVariance and Standard Deviation

• Measure of how different the values in a set of numbers are from each other

• Variance:

• Standard Deviation:

22 )(1

1xx

ns i

2)(1

1xx

ns i

Deviations from the meanVariance and Standard Deviation

• Data set: 2,3,5,6Calculation:

22 )(

1

1xx

ns i

83.133.3)(1

1 2

xxn

s i

0.44/)6532(/ nxx i

Value of X (X- ) (X- )2

2 -2 4 3 -1 1 5 1 1 6 2 4

∑=0 ∑=10

x x

33.3)14/(10)(1

1 22

xxn

s iVariance

Standard Deviation

Three normal distributions: mean=0 s2=1 s2=2 s2=0.5

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-3 -2 -1 0 1 2 3

0,1 0,2 0,0.5Central Tendency

mean=0

LeptokurticHomogenous

Narrow scatter

PlatykurticHeterogeneous

wide scatter

Mesokurtic

Example 2: FEV1 (litres) of 57 male medical students

Table: FEV1 (litres) of 57 male medical students 2.85 3.19 3.50 3.69 3.90 4.14 4.32 4.50 4.80 5.202.85 3.20 3.54 3.70 3.96 4.16 4.44 4.56 4.80 5.302.98 3.30 3.54 3.70 4.05 4.20 4.47 4.68 4.90 5.433.04 3.39 3.57 3.75 4.08 4.20 4.47 4.70 5.00 3.10 3.42 3.60 3.78 4.10 4.30 4.47 4.71 5.10 3.10 3.48 3.60 3.83 4.14 4.30 4.50 4.78 5.10

Example 2: FEV1 (litres) of 57 male medical students

Mean: 4.06 Variance: 0.45

SD: 0.67 Q1: 3.54

Q2 (Median): 4.10 Q3: 4.52

Percentile 5.16 Range: 2.85 to 5.43

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FEV1 (litre)

Fre

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The Meaning of Standard Deviation

• How the data are dispersed around mean• Mean ± 1 SD represent 68.3% of the

population• Mean ± 2 SD represent 96% of the population• Mean ± 3 SD represent 99.7% of the

population

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The Meaning of Standard Deviation

±SD % of Pop

1 68.3

1.96 95

2 95.5

2.58 99

3 99.71SD 1SD

34% 34%

2SD 48% 2SD 48%

Standard Error of Mean (SEM)

• How confident can we be that the sample mean represents the population mean µ?

• SEM=SD/– SEM must be much smaller than the SD

• mean ± 1.96*SD cover 95% of the data• mean ± 1.96*SEM cover 95% of the

population mean• SEM and SD are different!

n

Standard Error of Mean (SEM)

• Describing the scatter or spread of data, use SD• Estimate population parameters, use SEM

• Epidemiologic study, SEM• Clinical or laboratory research, SD

Summarizing Data - CalculatorPut DATA below:

Interval Size:

0102030405060708090

N:µ:σ:

Mean: 4.06 Variance: 0.45

SD: 0.67 Q1: 3.54

Q2 (Median): 4.10 Q3: 4.52

Percentile 5.16 Range: 2.85 to 5.43

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Ylim:

New Data

HISTOGRAM

POLYGON

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Example Data

RUN

ReDraw

Box-Plot• The box itself contains the middle 50% of the

data. The upper edge (hinge) of the box indicates the 75th percentile of the data set, and the lower hinge indicates the 25th percentile. The range of the middle two quartiles is known as the inter-quartile range.

• The line in the box indicates the median value of the data.

• The + indicate mean value• The ends of the vertical lines or "whiskers"

indicate the minimum and maximum data values, unless outliers are present in which case the whiskers extend to a maximum of 1.5 times the inter-quartile range.

• The points outside the ends of the whiskers are outliers or suspected outliers. 0

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Box Plot – Example 2

• FEV1 of 57 students Serum triglyceride measurements in cord blood from 282 babies

What you can get from a box-plot?

• Graphically display a variable's location and spread at a glance. [Q1, Q2 (median), Q3, interquartile range]

• Provide some indication of the data's symmetry and skewness.

• Unlike many other methods of data display, boxplots show outliers.

• By using a boxplot for each categorical variable side-by-side on the same graph, one quickly can compare data sets.

• One drawback of boxplots is that they tend to emphasize the tails of a distribution, which are the least certain points in the data set. They also hide many of the details of the distribution. Displaying histogram in conjunction with the boxplot helps

Transformations

-2 -1.5 -1 -0.5 0 0.5 10

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log(triglyceride)

frequ

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triglyceride

frequ

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LOG (triglyceride)triglyceride

Summarizing data

• Univariate – categorical variable– Frequency distributions– Bar Chart, Pie Chart

Summarizing data• Univariate – continuous variable

– Grouped frequency distributions– Polygon or histogram– Mean, Median, Mode, Percentile, Q1, Q2, Q3,

extreme values– Standard deviation, variance, range, interquartile

range– Box-Plot– Normality test statistics

Next lecture ( Lecture 2)

• Bivariate – one is categorical and the other is continuous variable– t-test– ANOVA

Lecture 3 – categorical data analysis

• Bivariate – both are categorical– Contingency tables– Chi-square test

• Response is categorical, predictors could be both types.– Logistical regression

Lecture 4 – Continuous response

• Correlation• Multiple linear regression

• Thanks.

• Question?