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Many statistics books begin by defining the different kinds of variables you might want to analyze. This scheme was developed by Stevens and published in 1946.
A categorical variable, also called a nominal variable, is for mutual exclusive, but not ordered, categories. For example, your study might compare five different genotypes. You can code the five genotypes with numbers if you want, but the order is arbitrary and any calculations (for example, computing an average) would be meaningless.
A…

Define Scaling:

refers to the construction of an instrument that associates qualitative constructs with quantitative units

scales - scaling - define

assignment of objects to numbers according to a rule.

How do scales work?

application of an instrument (scales) that associates qualitative constructs with quantitative units.

what are the 11 types of scales?

Thurstone

Guttman

Constant Sum

Likert

Line Marking

Pair Comparison

Q-Sort

Rank Order

Semantic Differential

Smiling Faces

Staples Scale

True/ False

Guttman

Constant Sum

Likert

Line Marking

Pair Comparison

Q-Sort

Rank Order

Semantic Differential

Smiling Faces

Staples Scale

True/ False

Define Thurstone scale

type of scale - also called equal appearing interval scaling. way of measuring people's attitude along a single dimension by asking if the person agrees or disagrees - with each os a large set of statements (~ 100) that are written about that attitude.

Based on the example, tell what type of scale: "people should exercise if they want to be healthy.

agree ------

disagree --------

or

true ------

false ------"

agree ------

disagree --------

or

true ------

false ------"

Thurstone scale. (Key words: Agree/ Disagree) *Know*

Define Guttman scale

a series of items of increasing intensity is ranked in order - from least to most extreme. Purpose to "establish a one-dimensional continuum for a concept you wish to measure." Focused on one concept.

Based on the example, assign the type of scale: "Place a check mark on the statement you agree with:

1. some children require physical restraint when unruly. (least extreme)

2. slapping a child's hand is an efficient discipline technique.

3. spanking is sometimes necessary to control children.

4. sometimes children require discipline with the belt or whip." (most extreme)

1. some children require physical restraint when unruly. (least extreme)

2. slapping a child's hand is an efficient discipline technique.

3. spanking is sometimes necessary to control children.

4. sometimes children require discipline with the belt or whip." (most extreme)

Guttman or cumulative scale. (Key words least to most extreme)

Define Paired comparison scale

look at different type of statements to choose between pairs - possible to identify areas more important to the participants. Key concept: pairs to choose from.

Based on the example, assign the type of scale: "4 areas of service improvement: frequency/ speed/ type/ price.

For each of the following pairs, choose which you believe it needs more improvement:

----- frequency/ speed

----- frequency/ type

----- service/ type

----- type / price"

For each of the following pairs, choose which you believe it needs more improvement:

----- frequency/ speed

----- frequency/ type

----- service/ type

----- type / price"

Paired comparison scale.

Define Ranked Order scale

this scale orders a list of items the participant should rank each item. (Key concept: ranking based on the items)

Based on the example, assign the type of scale: "consider the amount of time you spend on each item and order from 1 to 5 - rank 1 to the least amount of time to 5 meaning it takes more of your time:

___ distributor

____ pharmacy manager

___inventory coordinator

___ pharmacist

____ technician"

___ distributor

____ pharmacy manager

___inventory coordinator

___ pharmacist

____ technician"

Rank order scale

Based on the example, assign the type of scale: "My personal view of Bill:

least Neutral Most

characteristic characteristic

trait trait

-1 0 +1

friendly distant cold-hearted"

least Neutral Most

characteristic characteristic

trait trait

-1 0 +1

friendly distant cold-hearted"

Q-sort technique

Describe the Q-sort technique scale

each statement (~30 - 60 statements) expresses different opinions on a certain issue - statements are rank-ordered sort (most agree +1 to most disagree -1; most like +1 to most unlike -1, etc.) -- to measure characteristics/ attitudes of a person/ manager, politician, etc.

Define constant sum scale

allocate points to the criteria - divide 100 points among the choices in terms of the importance for each choice.

allocate more points to the criteria you find more important

allocate more points to the criteria you find more important

Given this example, determine what type of scale: "please divide 100 points among the following criteria in terms of their importance - allocate more points to the ones you consider more important:

___ ability to choose your own Dr.

___ price

___ speed of payment of claims

___ ability to choose the hospital"

___ ability to choose your own Dr.

___ price

___ speed of payment of claims

___ ability to choose the hospital"

Constant Sum Scale

Given the example, determine the type of scale:

"to what extent are the demand levels of your business products unstable/ unpredictable

not significant extent I________I to a greater extent"

"to what extent are the demand levels of your business products unstable/ unpredictable

not significant extent I________I to a greater extent"

line marking scale (pick the portion of the line that it's closer to the option you prefer

"Choose the option you prefer in regards the service:

;^) :^( :^S "

;^) :^( :^S "

smiling faces

the participant is provided with a range of possible responses to a number of positive and negative statements.

Likert scale

"people should exercise if they want to be healthy:

___ strongly disagree

___ disagree

___ agree

___ strongly agree"

___ strongly disagree

___ disagree

___ agree

___ strongly agree"

likert scale *KNOW* most common!

"reaction to TV commercial:

+3

+2

+1

exciting

-1

-2

-3"

+3

+2

+1

exciting

-1

-2

-3"

stapel scale

polar opposites within a set of paired polar opposite adjectives used ot describe values

semantic differential scale

"exciting ____________pilots_________ boring"

semantic differential scale

"honest__________mechanics_______ disonest"

semantic differential scale

"TV commercials are more exciting during the Superbowl:

___ strongly disagree

___ disagree

___ agree

___ strongly agree"

___ strongly disagree

___ disagree

___ agree

___ strongly agree"

Likert scale

match the definition with the scale type:

Increase intensity from lower to extreme

Increase intensity from lower to extreme

Guttman scale

match the definition with the scale type: strongly disagree - disagree - not agree nor disagree - agree - strongly agree

Likert Scale *Know*

-3 -2 -1 exciting +1 +2 +3

staple scale

true ( ) or false ( )

Thurstone scale *Know*

Constant Sum

Guttman

*Likert

Line Marking

Pair Comparison

Q-Sort

Rank Order

Semantic Differential

Staples Scale

Smiling Faces

*Thurstone

Guttman

*Likert

Line Marking

Pair Comparison

Q-Sort

Rank Order

Semantic Differential

Staples Scale

Smiling Faces

*Thurstone

*Thurstone*: True/ False; agree/ disagree

*Likert*: strongly disagree - disagree - nor agree or disagree agree strongly agree

Guttman: ↑intensity from least to most EXTREME

Line Marking -- mark preference on a line mark - choose from low to higher preference

Semantic Differential - opposite pairs with degrees to choose in between (honest - item to measure - disonest, etc.)

Staples Scale -- -3 -2 -1 exciting +1 +2 +3

Pair Comparison - 4 pairs with 4 paired-characteristics to pick one pair

Q-Sort -- preferences in PILES / categories

Rank Order - list of items to rank from 1-7, for ex.

Constant Sum: allocate points ↑more, ↑liking

Smiling Faces - ;^)

*Likert*: strongly disagree - disagree - nor agree or disagree agree strongly agree

Guttman: ↑intensity from least to most EXTREME

Line Marking -- mark preference on a line mark - choose from low to higher preference

Semantic Differential - opposite pairs with degrees to choose in between (honest - item to measure - disonest, etc.)

Staples Scale -- -3 -2 -1 exciting +1 +2 +3

Pair Comparison - 4 pairs with 4 paired-characteristics to pick one pair

Q-Sort -- preferences in PILES / categories

Rank Order - list of items to rank from 1-7, for ex.

Constant Sum: allocate points ↑more, ↑liking

Smiling Faces - ;^)

Descriptive statistics

Identify:

types of measurements scales,

types of variables;

organize, display and interpret data

types of measurements scales,

types of variables;

organize, display and interpret data

what are the types of Frequency distributions?

1. Absolute versus Relative

2. Simple versus Cumulative

2. Simple versus Cumulative

what is Absolute frequency?

The number of observations in a given statistical category. Ex.: experts investment would be the the individual and total sum of each category - Real Estate, Art, Precious Metals example.

What is the Relative frequency?

It's the number of the category ÷ total sum of all categories x 100 -→ result in % In the same 'experts' example, the higher % would be highest recommended and the lowest % would be the least recommended.

Exercise -- sample of investment experts was asked what would the money be invested in: Real Estate (R), Commodities (C), Arts (A), Stocks (S). Result: CCCSSSRRRARRR

Provide the best and worst investment based on the data.

Provide the best and worst investment based on the data.

Thus: Simple/ abs freq. Relative Frequency

C 3 3 x 100 : 13 = 23%

S 3 3 x 100 : 13 =23%

R 6 3 x 100 : 13 = 46%

A 1 3 x 100 : 13 =8%

Total 13

R is most recommended and A is the least recommended.

C 3 3 x 100 : 13 = 23%

S 3 3 x 100 : 13 =23%

R 6 3 x 100 : 13 = 46%

A 1 3 x 100 : 13 =8%

Total 13

R is most recommended and A is the least recommended.

1.) What are the 2 types of Continuous variables?

2.) Define Interval variables:

3.) Define Ratio variables:

4.) What is the mean?

5.) What is the median?

6.) What is the mode?

7.) Which is sensitive to outliers?

8.) Which is insensitive to outliers?

9.) Which is used for ordinal OR continuous data?

10.) Which is only used for continuous data?

2.) Define Interval variables:

3.) Define Ratio variables:

4.) What is the mean?

5.) What is the median?

6.) What is the mode?

7.) Which is sensitive to outliers?

8.) Which is insensitive to outliers?

9.) Which is used for ordinal OR continuous data?

10.) Which is only used for continuous data?

1.) Interval & Ratio

2.) Ranked in order (similar to ordinal) BUT with consistent change in magnitude between units

3.) "Like" interval, but w/ absolute zero (never ends)

4.) Average

5.) Middle number

6.) # the occurs most frequently

7.) Mean

8.) Median

9.) Median

10.) Mean (not sure why, but whatever...)

2.) Ranked in order (similar to ordinal) BUT with consistent change in magnitude between units

3.) "Like" interval, but w/ absolute zero (never ends)

4.) Average

5.) Middle number

6.) # the occurs most frequently

7.) Mean

8.) Median

9.) Median

10.) Mean (not sure why, but whatever...)

Statistical Applications in Pharmacy Settings:

1.) Does anyone have this packet in ENGLISH? Mine appears to have printed in another language...

2.) What is a Practice-Based Research Network (PBRN)?

3.) What are the 2 types of Discrete variables?

4.) What is a Nominal variable?

5.) What is an Ordinal variable?

1.) Does anyone have this packet in ENGLISH? Mine appears to have printed in another language...

2.) What is a Practice-Based Research Network (PBRN)?

3.) What are the 2 types of Discrete variables?

4.) What is a Nominal variable?

5.) What is an Ordinal variable?

1.) Group of practitioners (usually ambulatory care settings) that offer pharmacy residents research experience. (a group of practices which will let you study their patients, processes, etc... I think)

2.) Nominal and Ordinal

3.) Classified into groups in NO particular order and no indication of severity

4.) Classified into groups IN specific order, but no level of magnitude difference between ranks.

2.) Nominal and Ordinal

3.) Classified into groups in NO particular order and no indication of severity

4.) Classified into groups IN specific order, but no level of magnitude difference between ranks.

1.) What is SD?

2.) T or F? SD is useful for ordinal and continuous data.

I guess that's it. Thank the statistics gods!!!!!! Good

luck tomorrow. I hear its not bad. Make sure to look

over her examples!!

2.) T or F? SD is useful for ordinal and continuous data.

I guess that's it. Thank the statistics gods!!!!!! Good

luck tomorrow. I hear its not bad. Make sure to look

over her examples!!

1.) Standard ???distribution (I think this is that bell curve thingy...)

2.) False!! ONLY used for Continuous data

2.) False!! ONLY used for Continuous data

Scaling Types SUMMARY: **KNOW**

1.) Tell whether you strongly dislike, dislike, indifferent, like, or strongly like the following...

2.) Sort the statements according to which are most important to you and least important to you.

3.) Question: NSU works against, not for its students. Agree or disagree?

4.) Another name for Thurston Scale:

5.) Assume that those who agree w/ the extreme position also agree w/ the preceding less extreme positions.

6.) Question: What do you think about about the difficulty of pharmacy school? HARD (very - kinda - neither - kinda - very) EASY

7.) Q: What was your reaction to the Superbowl? Excitement level +3, +2. +1, 0, -1, -2, -3.

1.) Tell whether you strongly dislike, dislike, indifferent, like, or strongly like the following...

2.) Sort the statements according to which are most important to you and least important to you.

3.) Question: NSU works against, not for its students. Agree or disagree?

4.) Another name for Thurston Scale:

5.) Assume that those who agree w/ the extreme position also agree w/ the preceding less extreme positions.

6.) Question: What do you think about about the difficulty of pharmacy school? HARD (very - kinda - neither - kinda - very) EASY

7.) Q: What was your reaction to the Superbowl? Excitement level +3, +2. +1, 0, -1, -2, -3.

1.) Linkert scale

2.) Q-Sort scale

3.) Thurston scale

4.) Equal-Appearing Interval scale

5.) Guttman scale

6.) Semantic scale

7.) Staple scale

2.) Q-Sort scale

3.) Thurston scale

4.) Equal-Appearing Interval scale

5.) Guttman scale

6.) Semantic scale

7.) Staple scale

Scaling:

1.) T or F? There are many different types of scales including Thurston, Guttman, Q-sort technique, Likert, Staple and Semantic Differential scales.

2.) Which is referred to as "cumulative scaling" b/c it establishes a 1-dimensional continuum for concept you wish to measure. Focuses on only 1 concept.

3.) Which is a "way of measuring people's attitudes along a single dimension" by asking if they agree or disagree (ONLY 2 choices - agree or disagree)?

4.) Which can discriminate among a relatively large # of objects? (objects sorted into piles, piles ranked in order of preference)

5.) Which gives the participant a range of possible responses to a number of positive/negative statements?

6.) Which uses paired, polar opposite objectives to describe objects, attitudes, attributes, values, etc...?

1.) T or F? There are many different types of scales including Thurston, Guttman, Q-sort technique, Likert, Staple and Semantic Differential scales.

2.) Which is referred to as "cumulative scaling" b/c it establishes a 1-dimensional continuum for concept you wish to measure. Focuses on only 1 concept.

3.) Which is a "way of measuring people's attitudes along a single dimension" by asking if they agree or disagree (ONLY 2 choices - agree or disagree)?

4.) Which can discriminate among a relatively large # of objects? (objects sorted into piles, piles ranked in order of preference)

5.) Which gives the participant a range of possible responses to a number of positive/negative statements?

6.) Which uses paired, polar opposite objectives to describe objects, attitudes, attributes, values, etc...?

1.) True

2.) Guttman Scales (?)

3.) Thurston (Equal-Appearing Interval Scaling)

4.) Q-Sort Technique (items are then evaluated relative to each other)

5.) Likert scales (For example: completely disagree - disagree - don't care - agree - completely agree)

6.) Semantic Differential Scale (for example: Are pharmacists smart or dumb Dumb very - somewhat - neither - somewhat - very Smart)

2.) Guttman Scales (?)

3.) Thurston (Equal-Appearing Interval Scaling)

4.) Q-Sort Technique (items are then evaluated relative to each other)

5.) Likert scales (For example: completely disagree - disagree - don't care - agree - completely agree)

6.) Semantic Differential Scale (for example: Are pharmacists smart or dumb Dumb very - somewhat - neither - somewhat - very Smart)

Define scaling:

Construction of an instrument that associates qualitative constructs w/ quantitative units.

Principles for Table Construction:

- Don't try to do too much

- Use white space effectively

- Make sure tables and text refer to each other

- Not everything on the tables needs to be mentioned in the text

- Choose a criterion to order/group data [Size (smallest to largest), Chronology (first to last), Comparison]

- Use white space effectively

- Make sure tables and text refer to each other

- Not everything on the tables needs to be mentioned in the text

- Choose a criterion to order/group data [Size (smallest to largest), Chronology (first to last), Comparison]

4- Descriptive Statistics:

Measurement Scales

- Measurement: process of assigning numbers to characteristics according to defined rule.

- Hierarchy of measurement scales:

Measurement Scales

- Measurement: process of assigning numbers to characteristics according to defined rule.

- Hierarchy of measurement scales:

Hierarchy: NOIR

Define:

Nominal Scale

Nominal Scale

Nominal Scale: Classifies objects into categories based on some defined characteristics. Then the number of objects in each category is counted (i.e. count the number of observation with or without the attribute of interest). It is the least precise. Has no logical order of the categories. Categories are mutually exclusive (Ex: Gender, Religion, Eye color)

- Data on a nominal scale are called: Qualitative observations or Categorical observations

- Generally described in terms of: Percentages or Proportions

- Most commonly displayed in: Contingency tables or Bar graphs

- Data on a nominal scale are called: Qualitative observations or Categorical observations

- Generally described in terms of: Percentages or Proportions

- Most commonly displayed in: Contingency tables or Bar graphs

Ordinal Scale:

Ordinal Scale: It gives logical order to classification. Differences in the amount of the measured characteristic are discernible and numbers are assigned according to that amount. Categories are mutually exclusive. [Ex: letter-grading system (A, B, C, D, F)]

- Some observations have more or are greater than other observations.

- Generally described in terms of: Percentages or Proportions

- Most commonly displayed in: Contingency tables or Bar graphs

- Some observations have more or are greater than other observations.

- Generally described in terms of: Percentages or Proportions

- Most commonly displayed in: Contingency tables or Bar graphs

Interval Scale (i.e. Equal Unit Scale):

Interval Scale (i.e. Equal Unit Scale): Type of scale where the differences between levels of categories on any part of the scale reflect equal differences in characteristic measured. The point 0 is just another point on the scale. (Ex: temperature)

Ratio Scale:

Ratio Scale: Highest level in the hierarchy of measurement scales. Most precise measurement. A known/true zero point reflects an absence of the characteristic measured. Allows us to make statements about the proportional amounts of the characteristic that two objects possess. (Ex: a bag of apples weighing 30 lbs does weigh twice as much as one weighing 15 lbs.

Data and Variables

Raw Data:

Raw Data:

refers to the set of facts gathered by the statistician from the population or sample under investigation before anything has been done to the data

Variables:

- Qualitative Variables

- Qualitative Variables

Variables:

- Qualitative Variables - are measured on the nominal or ordinal scales, measurement consists of unordered or ordered (ranked) discrete categories

- Qualitative Variables - are measured on the nominal or ordinal scales, measurement consists of unordered or ordered (ranked) discrete categories

Variables:

- Quantitative Variables

- Quantitative Variables

- are measured on the interval or ratio scales.

- Types of Variables:

• Continuous with examples

• Continuous with examples

Variables falling in a certain interval on which no theoretical restrictions are placed. Measured along a scale (continuum).

- Examples: height, BP, time spent waiting in line, cholesterol, movie length, time, respiration, age, distance, temperature, annual snowfall amount, weight, battery life, improvement in SAT score, GPA

- Examples: height, BP, time spent waiting in line, cholesterol, movie length, time, respiration, age, distance, temperature, annual snowfall amount, weight, battery life, improvement in SAT score, GPA

- Types of Variables:

• discrete with examples

• discrete with examples

• Discrete = Have a restriction placed on them. There is NO continuity.

- Examples: gender, schooling level, day of the week, hair color, class (P1, P2, P3), race/ethnicity, college major, eye color, battery manufacturer, blood group (O, A, B, AB)

- Examples: gender, schooling level, day of the week, hair color, class (P1, P2, P3), race/ethnicity, college major, eye color, battery manufacturer, blood group (O, A, B, AB)

- Types of Variables:

• Dummy with examples

• Dummy with examples

• Dummy = Assumes a value of one if a criterion is met, a value of zero otherwise. (ex: Female = 1; Not Female = 0)

Descriptive Statistics

- Used to classify and summarize numerical data (i.e. to describe data)

- Provide simple summaries about the sample and the measures.

- Present quantitative descriptions in a manageable form

- Provide simple summaries about the sample and the measures.

- Present quantitative descriptions in a manageable form

Types of Statistical Analysis:

- Univariate: only one variable is involved

- Bivariate: the relationship between two variables is expressed.

- Multivariate: more than two variables are involved.

- Bivariate: the relationship between two variables is expressed.

- Multivariate: more than two variables are involved.

Major Characteristics of a Single Variable:

- Distribution

- Central Tendency

- Dispersion

- Central Tendency

- Dispersion

Frequency Distribution:

- Definition #1:

- Definition #1:

Frequency Distribution:

- Definition #1: It refers to a summary of the frequency of individual values or ranges of values for a variable.

- Definition #2: It is a tabulation that indicates the number of times a given score or group of scores occurs.

- Frequency distributions can be illustrates in: Tables and Graphs

- Definition #1: It refers to a summary of the frequency of individual values or ranges of values for a variable.

- Definition #2: It is a tabulation that indicates the number of times a given score or group of scores occurs.

- Frequency distributions can be illustrates in: Tables and Graphs

Rules for Developing Class Intervals for Frequency Distributions:

- There's no magic formula for the ideal width of class intervals.

- Use a minimum of 5 intervals and a maximum of 8 intervals. Sometimes it may be < 5 (Ex: gender)

- Whenever possible, the width of the class interval should be an odd number.

- Mutually exclusive (occurrence of one does not preclude occurrence of another); (Ex: 0-10, 11-19)

- All inclusive (ex: ≤ 10)

- Open-ended v. Close-ended - Examples of open-ended intervals = less than, above, over, under

- Number v. width (The more intervals the greater the precision, however it increases the complexity of the data.)

- Use a minimum of 5 intervals and a maximum of 8 intervals. Sometimes it may be < 5 (Ex: gender)

- Whenever possible, the width of the class interval should be an odd number.

- Mutually exclusive (occurrence of one does not preclude occurrence of another); (Ex: 0-10, 11-19)

- All inclusive (ex: ≤ 10)

- Open-ended v. Close-ended - Examples of open-ended intervals = less than, above, over, under

- Number v. width (The more intervals the greater the precision, however it increases the complexity of the data.)

Types of Frequency Distributions:

- Absolute Frequency - the number of observations in a given statistical category

- Relative Frequency - the ratio of the absolute frequency to the total number of data points in a frequency distribution

• Example: a class with a frequency of 6 within a sample size of 38 would have a relative frequency of:

6 38 = 0.158 x 100 = 15.8%

• Relative Frequency = (Absolute Frequency/Total # subjects) x 100

- Simple Frequency - how many numbers are in each class (i.e. category)

- Cumulative Frequency - It's constructed by adding the [absolute] frequency of scores in any class interval to the frequencies of all the class intervals below (or above) it on the scale of measurement. (i.e. It tells how many values are in that interval and all intervals less than it.

- Relative Frequency - the ratio of the absolute frequency to the total number of data points in a frequency distribution

• Example: a class with a frequency of 6 within a sample size of 38 would have a relative frequency of:

6 38 = 0.158 x 100 = 15.8%

• Relative Frequency = (Absolute Frequency/Total # subjects) x 100

- Simple Frequency - how many numbers are in each class (i.e. category)

- Cumulative Frequency - It's constructed by adding the [absolute] frequency of scores in any class interval to the frequencies of all the class intervals below (or above) it on the scale of measurement. (i.e. It tells how many values are in that interval and all intervals less than it.

Measure of Shape -- Shapes of Frequency Distributions:

- The shape depends on how the scores are distributed on the scale of measurement.

- Shapes: Uniform or Non-uniform, Skewed or Symmetric (symmetric may exhibit different degrees of kurtosis)

• Uniform or Rectangular Distribution - when all scores are evenly distributed throughout a distribution

- Shapes: Uniform or Non-uniform, Skewed or Symmetric (symmetric may exhibit different degrees of kurtosis)

• Uniform or Rectangular Distribution - when all scores are evenly distributed throughout a distribution

Shapes of Frequency Distributions:

• Skewness - refers to departures of a distribution from symmetry. In a negatively skewed distribution the tail of a distribution points toward the low scores. Distributions with a tail pointing toward high values of a variable are positively skewed.

Shapes of Frequency Distributions:

• Symmetric - when two halves of a graph would coincide if the graph were folded along a central line. The most familiar symmetric distribution is the bell-shaped distribution (known as the normal distribution).

Kurtosis = Degree of peakedness. The extent to which, for a given standard deviation, observations cluster around a central point. Degrees of kurtosis:

• Leptokurtic: appears taller and narrows

• Platykurtic: appears elongated and flat

• Mesokurtic: tends to be bell-shaped like the normal curve. It has a moderate peak representing a normal number of scores in the middle of the distribution. It appears symmetrical. A perfect mesokurtic curve = a normal curve.

Kurtosis = Degree of peakedness. The extent to which, for a given standard deviation, observations cluster around a central point. Degrees of kurtosis:

• Leptokurtic: appears taller and narrows

• Platykurtic: appears elongated and flat

• Mesokurtic: tends to be bell-shaped like the normal curve. It has a moderate peak representing a normal number of scores in the middle of the distribution. It appears symmetrical. A perfect mesokurtic curve = a normal curve.