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### CTRC Core Curriculum Seminar Series

Descriptive Statistics:

Data Types and Measures, Central Tendency, Variability

Chang-Xing Ma, PhD

Associate 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

- A random sample is such a representative sample

Some terminology

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

- Statisticis 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

(µw,σw2),

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

A random sample: sample size =

Gender Height Weight

statistics

A sample

mean height:

std height:

mean weight

std weight

% of male (=1)

Quantitative variables

Qualitative variables

Quantitative

continuous

Qualitative

nominal

Quantitative

discrete

Qualitative

ordinal

Data Types

- Numerical (Quantitative)
- numerical measurement
- Height
- Weight

- numerical measurement
- Categorical (Qualitative)
- with no natural sense of ordering
- Gender
- Hair color
- Blood type

- with no natural sense of ordering

Numerical Variable

- Continuous
- Range of values
- Height in inch

- Range of values
- Discrete
- Limited possible values
- # of smoking per day
- # of children in a family

- Limited possible values
- 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

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)

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

Code:

Race:

W – White

B – Black

H – Hispanic

A – Asian

Sex:

F – Female

M – Male

Poverty:

0 – not poor

1 – poor

Student ID Race Sex Reading Math Poverty

Descriptive Statistics

- Categorical variables:
- Frequency distribution
- Bar chart, pie chart
- Contingency tables

- Continuous variables:
- Grouped frequency table
- Central Tendency
- Variability

An ordered arrangement that shows the frequency of each level of a variable.

Simple Frequency Distributionrace 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

It is useful for categorical variable level of a 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

Simple Frequency DistributionBar Chart level of a variable.

BY

- 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

Pie Chart level of a variable.

- 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.

Complex frequency distribution Table level of a variable.

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

How about continuous variables? level of a variable.

- How data is distributed?
- Measure of Central Tendency
- Measure of Variability

Grouped Frequency Distribution – level of a variable.for continuous variable

Frequency Table

DATA:

Interval Size:

N:

µ:

σ:

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.

Grouped Frequency DistributionDescribing Distributions that it is

- Bell-Shaped Distribution
- Normal distribution N (µ=0, σ2 =1)
- t-distribution

µ, σ2

Probability density function - Normal that it is

z-transform

green curve is

standard normal

distribution

The that it is 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

Measure of Central TendencyMean, Median, ModeThe that it is 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=4.5

Measure of Central TendencyMean, Median, ModeSplit Data

The that it is Mode

is the midpoint of the interval that has highest frequency

robust to outlying value, but sometimes misleading

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

Mode=4, which occurred 3 times.

Measure of Central TendencyMean, Median, ModeMost frequently

Frequency distribution that it is

Histogram, Polygon graph

Bar Chart, Pie Chart

Describing Distributions

Mean, Median, Mode

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

Problem 2 that it is

- 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 ?

The sample range that it is

Interquartile range

The sample standard deviation (SD), variance

Standard error of mean (SEM)

Descriptive StatisticsVariabilityRange – the difference between the lowest and highest that it is For 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 - RangeMeasures of Dispersion - that it is Range

- All of curves have the same range
- Mean?
- Median?

Percentiles that it is : 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 DispersionPercentiles, Deciles, QuartilesInterquarile Range that it is – the 25 percentile (1st quartile) to 75 percentile (3rd quartile)

Age of Patients (years):2 3 3 4 5 66 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 RangeMeasures of Dispersion that it is Interquarile Range

Robust estimate, less efficient

Deviations from the mean that it is Variance and Standard Deviation

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

BUT

Deviations from the mean that it is Variance and Standard Deviation

- Measure of how different the values in a set of numbers are from each other
- Variance:
- Standard Deviation:

Deviations from the mean that it is Variance and Standard Deviation

- Data set: 2,3,5,6
Calculation:

Value of X (X- ) (X- )2 2 -2 4

3 -1 1

5 1 1

6 2 4

∑=0 ∑=10

Variance

Standard Deviation

Three normal distributions: mean=0 that it is s2=1 s2=2 s2=0.5

Leptokurtic

Homogenous

Narrow scatter

Mesokurtic

Platykurtic

Heterogeneous

wide scatter

Central Tendency mean=0

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

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 5.20

2.85 3.20 3.54 3.70 3.96 4.16 4.44 4.56 4.80 5.30

2.98 3.30 3.54 3.70 4.05 4.20 4.47 4.68 4.90 5.43

3.04 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

How the data are dispersed around mean that it is

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

The Meaning of Standard DeviationStandard Error of Mean (SEM) that it is

- 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!

Standard Error of Mean (SEM) that it is

- Describing the scatter or spread of data, use SD
- Estimate population parameters, use SEM
- Epidemiologic study, SEM
- Clinical or laboratory research, SD

Box-Plot that it is

- 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.

FEV1 of 57 students that it is

Box Plot – Example 2Serum triglyceride measurements in cord blood from 282 babies

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

What you can get from a box-plot?Univariate – categorical variable glance. [Q1, Q2 (median), Q3, interquartile range]

Frequency distributions

Bar Chart, Pie Chart

Summarizing dataUnivariate – continuous variable glance. [Q1, Q2 (median), Q3, interquartile range]

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

Summarizing dataBivariate – one is categorical and the other is continuous variable

t-test

ANOVA

Next lecture ( Lecture 2)Bivariate – both are categorical variable

Contingency tables

Chi-square test

Response is categorical, predictors could be both types.

Logistical regression

Lecture 3 – categorical data analysis- Thanks. variable
- Question?

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