2 introduction description

STATISTICS IN SPORT AND BEHAVIOURAL

SCIENCES SRT605

9/20/2011

SRT605 STATISTICS IN SPORT AND BEHAVIOURAL

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INTRODUCTION

&

DESCRIPTIVE

STATISTICS PN WAHIDAH TUMIJAN

BIOSTATISTICIAN

FAKULTI SAINS SUKAN DAN REKREASI

UiTM SHAH ALAM

SRT605 STATISTICS IN SPORT AND

BEHAVIOURAL SCIENCES 1

INTRODUCTION

TO

STATISTICS



What Is Statistics? • Statistics is the science of collecting,

describing and interpreting data to assist in making more effective decisions.

• Statistics provides a framework of handling data Important to know how to recognize data forms

and to use the correct statistical technique

• Using statistics, you will learn to correctly analyze your data to make inference and draw robust conclusions



Who uses statistics?

• Statistical techniques are used extensively by – Marketing, accounting, quality control,

consumers, professional sports people, hospital administrators, educators, politician, researchers, students, etc………..




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Basic Terms used….. • Population:

– A collection, or set, of individuals or objects or events whose properties are to be analyzed

• Sample: – A subset of a population – Represents the population by sampling

technique

• Parameter: – A numerical value summarizing all the data

of an entire population

• Statistic: – A numerical value summarizing the sample

data



Basic Terms used…..cont • Variable (or response variable):

– A characteristic of interest about each individual element of a pop or sample

• Data (singular):

– The value of the variable associated with one element of a pop or sample

– This may be a number, a word, or a symbol

• Data (plural):

– The set of values collected for the variable from each of the elements belonging to the sample



Classification of variables

Categorical

A variable that describes or categorizes an element of a

population

Nominal Ordinal

Numerical

A variable that quantifies an element of a population

Interval Ratio SRT605 STATISTICS IN SPORT AND


• Nominal variable: ▫ Classify characteristics of variables into categories

▫ Data is mutually exclusive (non overlapping) and not in rank order

▫ Eg: Gender, races

• dichotomous variable o Patient status

1 = alive

2 = death

o Blood Pressure

1 = high

2 = low



Categorical Variable


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• Ordinal variable: ▫ incorporates an ordered position, or ranking ▫ Difference/distances between ranks do not exist ▫ Mutually exclusive ▫ Eg: Socioeconomic Status

1 = Low 2 = Intermediate 3 = High

Attitude Scale 1 = Strongly Agree 2 = Agree 3 = Neutral 4 = Disagree 5 = Strongly Disagree



Categorical Variable…cont • Interval variable: ▫ A quantitative scales variables (discrete or

continuous) Discrete variable: A quantitative variable that can assume a countable

number of values (gaps between values)

Continuous variable: A quantitative variable that can assume an uncountable

number of values (with decimal values)

▫ Zero point is arbitrary ▫ Able to add or subtract ▫ Eg: Continuous variable (with decimal values) Temperature scale: 37.5ºC, 38.2ºC, etc

Discrete variable (gaps between values) No. of children : 1, 7, 10, …etc



Numerical Variable

• Ratio variable:

– Very similar to interval but zero point is not arbitrary

– Able to multiply or divide the values

– Eg:

• Temperature in Kelvin scale – 0 point is physically zero or no value

• Blood pressure – 120 mg/80 mg



Numerical Variable…cont Study Variable



1. Dependent or outcome variable: an outcome whose variation the study seek to describe, explain or account for by the influence of independent or explanatory variables.

2. Independent variable / explanatory variable: The independent/ explanatory variable that is hypothesised to influence the outcome variable under study; the hypothetical causal variable.

Eg: Difference between football player and

basketball player in relation to their leg power score.


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Classification of Statistics 1. Descriptive statistics

▫ Describe what happened in a particular study ▫ A collection, presentation, and description of

sample data ▫ Eg: table, graph , etc

2. Inferential statistics ▫ Draw conclusions about what those results

mean in some broader context ▫ Technique of interpreting the values resulting

from the descriptive techniques and making decisions and drawing conclusions about the population

▫ Allow researcher to generalize characteristics of a “pop” from the observed characteristics of a “sample”





Population

Sample

Sampling technique

Inferential statistics

Descriptive statistics

Table Graph

Frequency

Mean Median

Mode

Hypothesis

testing

Estimation



DESCRIPTIVE

STATISTICS

Organizing & Displaying data



Categorical variable

Frequency

Bar graph/chart

Pie chart

Numerical variable

Measure of Central Tendency

(mean, median, mode)

Measure of Variability

(variance, standard deviation, range, outlier)

Measure group position

(quartile, inter quartile range, percentiles and percentile ranks, standard score [Z-score])

Graphical presentation


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Categorical variable



Categorical variable FREQUENCY TABLES

• A listing, often expressed in chart form, that pairs each value of a variable with its frequency

• Table organized data into values and categories with titles and caption

• A frequency table may include: – Categories

– Frequency

– Cumulative frequency

– Relative frequency • Proportion (%)



Example of Frequency Table (SPSS output)



Gender

Frequency Percent Valid Percent Cumulative Percent

Valid Male 16 53.3 53.3 53.3

Female 14 46.7 46.7 100.0

Total 30 100.0 100.0

Step to generate Frequency Table from SPSS

Use speed driven.sav Open a data Analyze Descriptive Statistics Frequencies




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Open a data Analyze Descriptive Statistics

Frequencies



Open a data Analyze Descriptive Statistics

Frequencies

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• Show the amount of data that belongs to each category as proportionally sized rectangular areas

• Graphical presentation of frequency distribution of categorical data (nominal or ordinal)



Categorical variable BAR GRAPH / CHART



y axis: Frequency or

relative frequency

Height represents frequency or

percent

x axis: Categorical

variables Bars separate by

equal gaps

Bars of equal width

Type of Bar Charts



cluster

stacked

Step to generate Bar chart from SPSS

Use data dengue.sav

Open data Graphs Legacy Dialogs Bar



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OUTPUT • Show the amount of data that

belongs to each category as a proportional part of a circle

• Graphical presentation of frequency distribution of categorical data (usually nominal)



Categorical variable PIE CHART


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Each piece of slice represent each category

Size of slice represent frequency or percent

Use data dengue.sav

Open data Graphs Legacy Dialogs Pie



Step to generate Pie chart from SPSS

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OUTPUT

NO LABELING??



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Excellence Graph • The graph represents large data sets

concisely and coherently

• The ideas and concepts conveyed are clearly understood by the viewer

• The graph encourages the viewer to compare 2 or more variables

• The display induces the viewer to address the substance of the data and not the form of the graph

• There is no distortion of the data SRT605 STATISTICS IN SPORT AND


Numerical variable



Numerical variable

Measure of Central Tendency

(mean, median, mode)

Measure of Variability

(variance, standard deviation, range, outlier)

Measure group position

(quartile, inter quartile range, percentiles and percentile ranks, standard score [Z-score])

Graphical presentation


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1. Mean

Sample average Formula:

= sample mean = summation of all the x value

n = sample size

Sensitive to extreme value, where 1 data point could

make a great change in sample mean



Numerical variable MEASURES OF CENTRAL TENDENCY Example:

• What is the mean of systolic blood pressure (SBP) among the cases below? x1 = 120 x2 = 80 x3 = 90

x4 = 100 x5 = 120 x6 = 110

Solution:

FORMULA: n = 6 = 120+80+90+100+120+110 = 620

= 620 / 6

= 103.33



2. Median

Middle value or the 50th percentile of a set of ordered numbers / measurement

When n is odd, the middle value = [(n+1)/2]th

When n is even, median is the average of two middle most observation

Median = mean in normally distributed data

Not sensitive to extreme values



Example: a) n is odd:

In the opening round of the Christmas basketball tournament, Slippery Ice went into a freeze in the final 5 minutes of the game to preserve a 68-64 victory over a very tough and talented team from Hard Rock College. The starting five for Slippery Ice scored 12, 7, 18, 9 and 6 points. Find the median.

SOLUTION:

i- arrange the observations in order

6 7 9 12 18

n=5

Formula: median = [(n+1)/2]th

= [(5+1)/2]th

= 3th

median = 9 SRT605 STATISTICS IN SPORT AND



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b) n is even:

A new brand of cigarettes called Wheeze has just become available to

the public. The nicotine contents for a random sample of 6 of these cigarettes are 12.3, 18.1, 15.7, 16.9, 21.2, and 18.5 milligrams. Find the median.

SOLUTION:

i- arrange the observations in order

12.3 15.7 16.9 18.1 18.5 21.2

Median = middle of the observation

= (16.9 + 18.1) / 2

= 17.50 miligrams



Example: 3. Mode

Value which occurs most often or with the greatest frequency

Less useful in describing statistics

It requires no calculation

It can be more than 1

Example: ◊ The donation received from the residents of Windsor Lake

toward the American Cancer Society were recorded as follows: 3,4,5,6,7,7,7,7,8,8 and 9 dollars. Find the mode.

SOLUTION:

mode = occurs most often

= 7 (4 times)



Hands-on:

Find mean, median and mode from

data below :

x1 = 10 x2 = 8 x3 = 7 x4 = 15

x5 = 10 x6 = 10 x7 = 12 x8 = 13

x9 = 9 x10 = 7





Find mean, median and mode from

data below :

x1 = 10 x2 = 8 x3 = 7 x4 = 15

x5 = 10 x6 = 10 x7 = 12 x8 = 13

x9 = 9 x10 = 7 x11 = 14 x12 = 11

x13 = 7 x14 = 9 x15 = 9


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1) Variance

¤ Considers the position of each observation relative to the mean of the set.

¤ Measures the amount of spread or variability of observation from mean

¤ Formula:

variance,



Numerical variable MEASURES OF VARIABILITY Example:

1. An inventory of office equipment in 4 randomly selected departments showed that physics is in possession of 2 calculators, chemistry has 5, mathematics 7, and business administration 10. find the variance.

SOLUTION:

i- find the mean, = (2 + 5 + 7 + 10 ) / 4 = 6

ii- formula variance,

= (2 - 6)2 + (5 - 6)2 + (7 - 6)2 + (10 - 6)2

4 – 1

= 34 / 3

= 11.333



Example: 2. Find the variance for the value in the table below.

SOLUTION:

find the mean, = (2.3 + 3.4 + 2.6 + 1.8 + 2.9 + 3.1 ) / 6

= 16.1 / 6

= 2.7

variance, s2 = 1.67 / (6-1)

= 0.3344 SRT605 STATISTICS IN SPORT AND


x

2.3 2.7 0.16

3.4 2.7 0.49

2.6 2.7 0.01

1.8 2.7 0.81

2.9 2.7 0.04

3.1 2.7 0.16

= 1.67

2) Standard deviation

¤ Square root of variance

¤ Most widely used and better measure of variability

¤ The smaller the value, the closer to the mean

¤ Sensitive to extreme value

¤ Formula:

standard deviation,




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Example: i. Jack Knife was randomly given the following scores by 6 judges on his first dive at

the Hunting Hills annual swim meet: 4,5,6,6,7, and 8. Find the standard deviation.

SOLUTION:

i- find the mean, = (4+5+6+6+7+8) / 6

= 6

ii- formula standard deviation,

= √10/5

= 1.41



Example: ii. Find the standard deviation for the observation below.

SOLUTION:

i-find the mean, = (6+5+7+9+8+5+6) / 7

= 46 / 7

= 6.57

ii-formula standard deviation,

= √13.71/6 = 1.51 SRT605 STATISTICS IN SPORT AND


x

6 6.57 0.32

5 6.57 2.46

7 6.57 0.18

9 6.57 5.90

8 6.57 2.04

5 6.57 2.46

6 6.57 0.32

= 13.71

Hands on

• 20

• 15

• 17

• 19

• 18

• 15

• 16



Find variance and standard deviation

3) Range: « Simplest and least useful measure of variability o Only for quick estimate of variability

« The differences between the maximum and minimum value of the distribution

« Tends to increase with sample size « Sensitive to very extreme values

« Eg: The ages of the 5 children in the cast of the new musical opening at the Star City

Playhouse are 8,12,12,15, and 17 years, Find the range.

SOLUTION: The range of the 5 ages is #max – min= 17 – 8 = 9 years




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4) Outlier

« Also call extreme values

« Values that are very small or very large relative to the majority of the values in a data set



Example: • Table below lists the 1995 population (in thousands) for the five Pacific states.

Notice that the population of California is very large compared to the populations of the other 4 states. Hence, it is an outlier. Show how the inclusion of the outlier affects the value of the mean.

SOLUTION:

i-find the mean without include California, = (5431+3141+604+1187) / 4

= 2590.75 thousand.

ii- find the mean with California, = (5431+3141+604+1187+31589) / 5

= 8390.4 thousand

*Including California causes more than a threefold increase in the value of the mean, as it changes from 2590.75 thousand to 8390.4 thousand.



State Population (thousands)

Washington 5431

Oregon 3141

Alaska 604

Hawaii 1187

California 31589

Hands on



School Number of students in Standard 1

SMK A 230

SMK B 345

SMK C 257

SMK D 307

Q1: Find mean and SD?




SMK A 230

SMK B 345

SMK C 659

SMK D 307



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SMK A 230

SMK B 345

SMK C 95

SMK D 307


§ What is the different between Q1 and Q2?

mean??

SD??

§ What is the different between Q1 and Q3?

mean??

SD??



What is your conclusion??

Outlier increase:

Outlier decrease:



1) Quartiles

§ 3 summary measures that divide a ranked dataset into 4 equal parts

≈ First quartile, Q1 :value of the middle term among the observations that are less than the median

≈ Second quartile, Q2 : same as the median of a data set

≈ Third quartile, Q3 : value of the middle term among the observations that are greater than the median



Numerical variable MEASURES GROUP POSITION

25% 25% 25% 25%

Q1 Q2 Q3


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2) Inter Quartile Range (IQR)

§ The difference between the third and the first quartiles

IQR = Q3 – Q1



Example: • The following are the scores of 12 students in a mathematics class.

75 80 68 53 99 58 76 73 85 88 91 79

a) Find the values of the 3 quartile.

b) Find the inter quartile range (IQR).

SOLUTION:

(a) First, rank the given scores in increasing order. Then, calculate the quartile

53 58 68 73 75 76 79 80 85 88 91 99

Q1 = (68+73)/2 Q2 = (76+79)/2 Q3 = (85+88)/2

= 70.5 = 77.5 = 86.5 SRT605 STATISTICS IN SPORT AND


Values less than the median Values greater than the median

Step using SPSS



SPSS output



Descriptives

Statistic Std. Error

Age (years) Mean 56.39 .384

95% Confidence Interval for Mean

Lower Bound 55.64

Upper Bound 57.14

5% Trimmed Mean 56.33

Median 56.00

Variance 177.626

Std. Deviation 13.328

Minimum 22

Maximum 88

Range 66

Interquartile Range 21

Skewness .049 .070

Kurtosis -.765 .141


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3) Standard score (Z-score)

☺ Also called z-score

☺ Position of a particular value of x has relative to the mean, measured in standard deviation

☺ Used to help make a comparison of 2 raw scores that come from separate populations

☺ Formula:



s

xxz

Example:

SOLUTION:

a: x = 45 = 38 s = 7

z score a = (45 – 38) / 7 = 1

b: x = 72 = 65 s = 14

z score b = (72 – 65) / 14 = 0.5

* conclusion: from the result, your score is one standard deviation above the mean, but your friend’s score is only half of a standard deviation above the mean. We can conclude that your score is slightly better than your friend’s score



x

x

You want to compare your Math score with your friend’s Math score from difference

class. Your score was 45 and your friend’s score was 72points. But mean score in your

class was 38points compare 65points in your friend’s class. However the standard

deviation on your class was 7 compare to 14 in your friend’s class. Which score are

more better?

• It also uses in standard normal distribution

• Standard normal distribution is the normal distribution of the standard variable z

• Properties of the standard normal distribution – Total area under the normal curve is equal to 1

– Distribution is mounded and symmetric

– Has a mean of 0 and standard deviation of 1

– The mean divides the area in half, 0.50 on each side

– Nearly all the area is between z=-3.00 and z=3.00



z

Example: Find the area under the standard normal curve between z=0 and z=1.52

SOLUTION

By using table Standard Normal Distribution below, z=1.52 located raw

labeled 1.5 and column label 0.02, at their intersection is 0.4357, the measure

of area or probability for the interval z=0.00 and z=1.52.

*Area or probability expressed as: P(0.00<z<1.52) = 0.4357



Z=0 Z=1.52


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• Graph are the visual presentation of frequency distribution and may show – Differences in spread (variability) – Difference in shape of the distribution

• Type of useful graphs:

– Histogram – Polygon – Stem and leaf – Line graph – Box plot – Scatter plot (correlation technique) SRT605 STATISTICS IN SPORT AND


Numerical variable GRAPHICAL PRESENTATION

Histogram



Each bar represent the interval class

Normality curve line

Bar height represent frequency or percent

Interval class, no gaps in between

Step to generate histogram using SPSS

Use data breast cancer survival.sav

Open data Graphs Legacy Dialogs Histogram



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OUTPUT

Polygon

• A graph that displays the data using lines to connect points plotted for the frequencies

• The frequencies represent the heights of the vertical bars in the histogram



The line segments pass through the

mid points at the top of the rectangles

The polygon is tied down at both ends

Stem and leaf plot

• Another tool for visually displaying continuous data

• Very similar to a histogram

• Allow for the easier identification of individual values in the simple

• Each numerical value is divided into 2 parts:

– The leading digit becomes the STEM

– The trailing digit becomes the LEAF



Example: Let’s construct a stem-and-leaf display for the 19 exam score below:

74 82 96 66 76 78 72 52 68 86 84 62 76 78 92 82 74 88

SOLUTION

i- At a quick, we can see scores in 50s, 60s, 70s, 80, and 90s.

ii- Display in vertical position and place the stem

iii- Places leaf for each stem and continues until end of all 19 score.

iv- Result for stem-and-leaf display as below.

5 | 2

6 | 6 8 2

7 | 6 4 6 8 2 6 8 4

8 | 2 6 4 2 8

9 | 6 2 SRT605 STATISTICS IN SPORT AND

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leaf


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Hands on

• 75 80 68 53 99 58 76 73 85 88 91 79 55 75 66 75 78 90



Step using SPSS Use data Breast Cancer Survival.sav

Open data Analyze Descriptive Statistics Explore



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OUTPUT

Box plot

• A graphical display that use descriptive statistics based on percentile

• Also called “5 number summary plot” : min, max,Q1,Q2 and Q3

• Provide information about central tendency and the variability of the middle 50% of the distribution

– The box represent the IQR, 25th to 75th percentile

– Outlier observations is 1.5 times the IQR away from the edges of the box (>3.0 times is extreme values)

– Smallest and largest values that make up the lines are the nearest values outside the outliers

• Can easily comparing continuous data in multiple groups : can plotted side by side





Outlier

The 50th

Percentile

(median)

Largest value

which is not

Outlier (max)

Smallest value

which is not

Outlier (min)

The 75th

percentile

The 25th

percentile

The

whiskers


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Scatter plot

• All the ordered pairs of bivariate data on a coordinate axis system

• The input variable, x is plotted on the horizontal axis

• The output variable, y is plotted on the vertical axis

• Can be use as a basic graphical presentation in correlation





Fit line and R2 important

in correlation

Thank you….



2 introduction description

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