PSYCHSTATS: MODULE 3

Descriptive Statistics

• involves methods of organizing, picturing, illustrating, and summarizing data and information derived from samples or populations

• Unlike inferential statistics, the goal of performing this is simply to create a profile of the dataset without drawing further conclusions.

Distribution

• set of test scores arrayed for recording or study

Raw Score

• straightforward, unmodified (often numerical) account of performance

Simple Frequency Distribution

• scores are listed alongside the number of times that each score has occurred

Grouped Frequency Distribution

• test score (class) intervals replace actual scores

Class Interval

(Max Value - Min Value) / Number of Levels Set

Histograms

• graph with vertical lines drawn at the true limits of each test score / class interval, thus forming a series of contiguous rectangles

• Ideal for continuous data

Frequency Polygon

• a continuous line connects various points where test scores / class intervals (x-axis) meet frequencies (y-axis)

• Ideal for continuous data

Bar Graph

• numbers indicative of frequency appear on the y-axis while reference to some categorization or label runs on the x-axis

• These graphs tend to be more useful for visualizing nominal or ordinal data.

Graphs

• may also be used to summarize data.

Measures of Central Tendency

• indicate average / midmost score between extremes in a distribution

Mean (̄x, M)

• sum of the frequencies divided by the number of cases

• the most common measure of central tendency is the arithmetic _____ , colloquially referred to as the “average.”

• This accounts the actual numerical value of every score within a dataset.

• Often the most apt measure of central tendency for continuous data when the distributions are believed to be approximately normal.

x̄

• represents the sample mean

μ

• represents the population mean.

x

• raw score in a set of score

Mean Solution

x̄ = ∑ x / N

Median

• The middle score in a distribution

• This is determined by ordering scores by magnitude in a list according to ascending or descending order.

• x͂, Md, Mdn, and Med are symbols used to represent this.

The median may be particularly useful when:

• A distribution is skewed or has a few extreme scores

• An individual has an unknown or undetermined score

• A distribution is said to be open-ended

• Scores are measured on an ordinal scale

Position of Median

N + 1 / 2

Mode

• Most frequently occurring score in a distribution

• Scores tied for “most frequently recurring” can create more than one _____ and are called multimodal distributions (e.g. bimodal = two ____) as they co-occur with the same frequency.

• There is no universal symbol for mode, so we can use Mo or Mod

The mode may be particularly useful when:

• Data are measured on a nominal scale

• Variables produce discrete data

• We indicate the shape of the distribution (peak)

Measures of Variability / Dispersion

• indicate how scores in a distribution are scattered or dispersed

Range

• difference between minimum and maximum scores

• distance covered by scores in a distribution

• simplest measure of variability to calculate, but potential use is limited

Range Formula

Xmax - Xmin

Interquartile Range (IQR)

• difference between Q1 and Q3 of a distribution

• range of scores comprising the middle 50% of a distribution

• based on quartiles, data points that divide the distribution into four equal sections or quarters (25% of the distribution).

• used when measuring central tendency with the median

Standard Deviation and Variance

• The ________ provides a measure of the standard or average distance from the mean, and it describes whether the scores are clustered closely or widely scattered around the mean

• While this as a concept is straightforward, actual equations are often more complex and lead to related concepts of variance before the standard deviation.

• The process of squaring deviation scores gets rid of plus and minus signs, but it also creates a measure of variability based on squared distances which are not very useful or easy to appreciate for descriptives

Deviation Scores

• The first step in finding the standard distance from the mean is to determine the ______ or distance from the mean, for each individual score.

• The _____ is the difference between a data point (score) and the mean.

• There are two parts to this: the sign (telling us if the score is above or below the mean) and the number (giving us the actual distance).

deviation for sample

• x – x̄

deviation for population

• x – μ

Variance

• equals the mean of the squared deviations and is the average squared distance from the mean

Standard Deviation

• we can take the square root of the variance and produce a more meaningful number

• is the square root of the variance. Conversely, the variance is the square of the _______________

Skewness and Kurtosis

• Using frequency polygons can help visualize the various shapes and forms taken by every distribution.

• Some distributions are symmetrical while others contain extreme scores in either direction.

• Some distributions are said to be ____ (extreme cases in one direction that the other) while variations in symmetrical distributions may also differ in terms of peakedness (_____)

Skewness

• Nature and extent to which symmetry is absent

Normal or no skew

• if equal on both sides

Positive skew

• when relatively few of the scores fall at the high end of the distribution (i.e. most scores are low)

Negative skew

• when relatively few of the scores fall at the low end of the distribution (i.e. most scores are high).

Perfectly Symmetrical or Normally Distributed

• Ssk = 0

Moderately Symmetrical

• –0.5 < Ssk < +0.5

Moderately Skewed

• ±0.5 < Ssk < ±1.0

Highly Skewed

• Ssk < –1.0 ; Ssk > +1.0

Kurtosis

• Refers to distribution steepness at its center

• The root suffix kurtic (curve / bulge)

Leptokurtic

• (from ___-, i.e. thin) has a tall peak or slender curve as well as fat tails

• κ = positive values

Platykurtic

• (from ____, i.e. flat) has a low peak or broad curve as well as thin tails

• κ = negative values

Mesokurtic

• (from _____, i.e. middle) has an intermediate peak as well as moderate tails

• “tails neither thin nor fat”

• κ = value of zero

Measures of Position / Location

• determines the position of a single value in relation to other values within a given sample or population dataset

Percentiles

• split sorted data into one hundred equal parts

• divides dataset into a hundred equal parts (per cent = per a hundred; -ile = point)

• Percentiles, quartiles, and deciles share some overlapping equivalences

  • Q1 = P25 ; Q2 = P50 ; Q3 = P75

  • D1 = P10 ; D2 = P20 ; D3 = P30 ; D4 = P40 ; and so on

• The ____ rank (PR) is the percentage of scores equal to or below the data point in question

Quartiles

• split sorted data into four equal parts

• divides dataset into four equal parts (per quart = per four ; -ile = point); not to be confused with quarters which are the sections between quartiles

• Percentiles, quartiles, and deciles share some overlapping equivalences

  • Q1 = P25 ; Q2 = P50 ; Q3 = P75 ; Q2 = D5

• The median is always conveniently located at the middle of a dataset, making it the 2nd quartile, the 5th decile, and the 50th percentile in distributions

Deciles

• split sorted data into ten equal parts

• Percentiles, quartiles, and deciles share some overlapping equivalences

  • Q2 = D5

  • D1 = P10 ; D2 = P20 ; D3 = P40 ; D5 = P50 ; and so on

z-Scores

• A raw dataset on its own can be difficult to work with, so we need a reference point or standard score to make it more meaningful.

• Finding distance from the mean yields a value called a ________ or standard score, indicates the direction and degree that any given raw score deviates from the mean of a distribution on a scale of units.

z

• represents a coordinate on the standard normal distribution (also known as the Z-distribution).

The Normal Curve

• sometimes called the Laplace-Gaussian curve or Gaussian curve

• bell-shaped, smooth, symmetrically defined curve

• tapers on both sides toward the x-axis asymptotically

Karl Pearson

• first to use the term “normal curve”

The Area Under The Normal Curve

• divided into areas defined in standard deviation units

• tail : area between 2 - 3 SD units

• simplifies score interpretation

• work well with standard scores

Basic Standard Scores

• raw scores converted from one scale to another with an arbitrarily set M and SD

z Score

• zero plus or minus one

• M = 0 ; SD = 1

• The _____ is the common standard score used as it matches the unit distance of the standard deviation. In a normal distribution, this score ranges from -4 to +4

T Score

• fifty plus or minus ten

• M = 50 ; SD = 10

• Used in psychological statistics and assessment. In a normal distribution, its range goes from 10 to 90.

Stanine

• standard nine

• M = 5 ; SD = ~2

Deviation IQ

• scores on IQ tests

• M = 100 ; SD = 15 *

Standard Score

• Finding distance from the mean yields a value called a ________ that indicates the direction and degree that any given raw score deviates from the mean of a distribution on a scale of units.

z score equation

(x - μ) / σ

t score equation

50 + ( z * 10)