Psychological Statistics Chapters 1-6
Population The Set of all individuals interested in a particular study Sample the set of individuals selected from a population usually intended to represent the population in a research study Variable a characteristic or condition that changes or has different values for different individuals AdvertisementUpgrade to remove ads Data measurement of observations data set a collection of measurements Datum a single measurement or observation called score or raw score Parameter a value usually numerical that describes a population Statistic A value, usually numerical that describes a sample Descriptive Statistics statistical procedures used to summarize or organize data Inferential Statistics consist of techniques that allow us to study samples then make generalizations about the population sampling error a naturally occurring discrepancy or error that exists between a sample statistic and the corresponding population parameter Correlational method two different variables are observed to determine whether there is a relationship between them limitations can demonstrate if there is a relationship between two variables but cannot explain why Experimental method one variable is manipulated while another variable is observed and measured. manipulation where researcher manipulates one variable Control where researcher controls one situation Participant variables categories such as age, gender, and intelligence that vary from one individual to another Environmental variables characteristics of the environment such as lighting, time of day, and weather conditions Independent variable the variable that is manipulated by the researcher Dependent variable the variable that is observed to asses the effect of the treatment Control condition do not receive experimental treatment Experimental Condition do receive experimental treatment Quasi-independent variable in a non experimental study the independent variable used to create different groups Constructs internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior Operational definition Identifies a measurement procedure for measuring an external behavior and uses the resulting measurements as a definition and a measurement of a hypothetical construct Discrete variable consists of separate and indivisible categories Continuous variable there are an infinite number of possible values that fall between any two observed values Real limits are bounders of intervals for scores that are represented on a continuous number line Nominal scale categories that can't be ranked Ordinal scale have names and can be ranked interval scale has real numbers that are divisible but no real zero point ratio scale has real divisible numbers and a 0 point Sigma adding everything together
chapter 1
raw scores scores that have not been averaged, sorted, or processed yet distribution is the pattern of a set of numbers frequency table A table for organizing a set of data that shows the number of times each item or number appears. AdvertisementUpgrade to remove ads cumulative percentages Also known as percentile. It is the percentage of scores equal to or lower than a given score value. cum % = crf · 100 grouped frequency table is a visual depiction of data that reports the frequencies within a given interval rather than the frequencies for a specific value histogram a bar graph that shows the frequency of data within equal intervals frequency polygon a graph line wih the x-axis representing values and the y-axis representing frequencies central tendency measure of the "central" scores in a data set, or where the group tends to cluster mean an average of n numbers computed by adding some function of the numbers and dividing by some function of n statistics the collection and classification of data that are in the form of numbers parameters the numbers based on the whole population median The middle number in a set of numbers that are listed in order mode the most frequent value of a random variable unimodal having a single mode bimodal distributions with two modes multimodal distributions with more than two modes outlier an extreme deviation from the mean range a measure of variability calculated by subtracting the lowest score from the highest score standard deviation the typical amount that the scores in a samoke vary, or deviate, from the mean deviation from the mean the amount that a score in a sample differs from the mean of the sample variance the average of the squared deviations from the mean sum of squares symbolizes as SS in the sum of the squared deviations from the mean fro each score normal distribution a specific frequency distribution in the shape of a bell-shaped, symmetric, unimodal curve skewness describes how much one of the tail of the distribution is pulled away from the center kurtosis describes how flat or how peaked a curve is positively skewed data, the distribution's tail extends to the right, in a positive direction floor effect a situation in which a constraint prevents a variable from taking values below a certain point ceiling effect a situation in which a constraint prevents a variable from taking values above a certain point negative skewed data, the distribution's tail extends to the left, in a negative direction mesokurtic describes a normal distribution leptokurtic describes curves that are tall and thin with thicker tails platykurtic describes curves that are short and fat with thinner tails interquartile range the difference between the first and third quartiles of a data set first quartile marks the 25th percentile of a data set third quartile marks the 75th percentile of a data set
chapter 2
What are the 3 most important components of a distribution 1. Center of a distribution (measures of central tendency) 2.The spread of a distribution (Measures of dispersion) 3. The shape of a distribution What are the three measures of central tendency mean, median mode What is central tendency What is typical/how people tend to respond AdvertisementUpgrade to remove ads When do you use english letters and when do you use greek letters greek letters= population english letters= sample Describe the mean - Arithmetic average - Add all scores and divide by the number of scores - Population symbol: μ ("mu") Sample symbol: M or x̅ ("x-bar") Describe the median -Score that represents the 50th percentile i.e., the exact (mathematical) middle -less influenced by outliers than the mean i.e., more reliable/robust - Place scores in order and find the middle Describe the mode -Value with the highest frequency -Unreliable, but the only measure of central tendency for nominal scales -Can be bimodal: has two subgroups, each with a meaningful mode what is the range - Range: highest score - lowest score (measure of dispersion) - Indicates the distance of the typical scores from the median - More reliable than the range, and less susceptible to extreme scores What is the SIQR -Semi Interquartile Range -Not affected by outliers -Used in conjunction with the median - (Q3-Q1)/2 what is the deviation score how far a score is from the mean (x-M) What is the deviation mean taking the absolute value of deviation score avg what is the zero sum principle the deviation score is always 0 SS SS: Take deviation scores, square them and add them up variance Variance: Take SS and divide by degrees of freedom standard deviation Standard deviation: Take the square root of the variance Describe a positive distribution -Positive skew -Scores are concentrated on the left (the lower limit) -Skewer POINTS TO THE RIGHT (towards the positive values) -Floor effect -Mean is greater than the median Describe a negative distribution -Scores are concentrated on the right (the upper limit) -Skewer POINTS TO THE LEFT (towards the negative values) -Ceiling effect describe floor effect Associated with positive skew; can't score any lower describe the ceiling effect Associated with a negative skew; can't score any higher why are the median and the SQIR associated they both deal with percentiles and are not affected by outliers Why are the mean and standard deviation associated they both give information about the variation the data and the avg. true or false: the mean is generally pulled in the direction of the skew true describe degrees of freedom -the number of scores there are free to vary (N-1) -used to correct Bias: Variance of a sample tends to underestimate the variance of a population What happens to the mean and SD by Adding/subtracting a constant (C) to every score in a distribution Mean: adds or subtracts that same value SD: not affected What happens to the mean and SD when you Multiple/divide each score by C (constant) Mean: multiplied or divided by that value SD: multiplied or divided by that value True or False: The standard deviation from the mean will be smaller than the standard deviation from any other point in the distribution. TRUE True or false: The sum of the squared deviations from the mean will be less than the sum of squared deviations around any other point in the distribution. TRUE What is the h spread of a Box and Whisker plot the interquartile range (Q3-Q1) What are the inner fences of a box and whisker plot anything outside the lower and upper fence = outliers What are adjacent values of a box and whisker plot the top and bottom of whiskers; they are the most extreme values with in upper and lower fences What is trimming (in regards to a box and whisker plot) deleting scores What is Winsorizing (in regards to a box and whisker plot) ? replacing outlying scores (highest and lowest); creates a trimmed mean and can skew stats
chapter 3
What are the 3 features present in all distribution of scores? -form -central tendency -variability These are independent of each other, and knowledge of all 3 features leads you to a fairly complete understanding of a distribution variability having more than one value; there are 4 measures of variability What are the 4 measures of variability? range, interquartile range (IQR), standard deviation, & variance AdvertisementUpgrade to remove ads range the highest score minus the lowest score Range = Xh - Xl where Xh=highest score & Xl=lowest score What does a large range of values mean? that there is too much variability in a process and adjustments are called for interquartile range the range of scores that contain the middle 50 percent of a distribution; it is an important element of boxplots boxplot a graphic that conveys the data of a distribution and some of its statistical characteristics with one picture How do you first go about finding the interquartile range? to do so, you must have the 25th percentile score and the 75th percentile score What is a percentile? the point below which a specified percentage of the distribution falls in terms of percentiles, what does the median do? it is the pt that divides a distribution into equal halves; therefore, the median is the 50th percentile How do you find the 25th and 75th percentile scores? The 25th percentile score is the one that has 25% of the scores below it; to find the 25th percentile score, multiply 0.25 times N (the number of scores); the value you get will be how many #s up from the bottom of the data set... this is the score of the 25th percentile the 75th percentile is found the same way except you do 0.75xN and the value you get will be how many #s down from the top of the data set Once you have the 25th and 75th percentile scores, what else do you do to find the interquartile range? the IQR is the 75th percentile minus the 25th percentile: IQR=75th percentile - 25th percentile: What's another way to find the interquartile range? find the midpoint of the data set; then take the midpt of the 1st half of the data set (this pt represents the 75th percentile); then take the midpt of the bottom half of the data set (this pt represents the 25th percentile) standard deviation a descriptive measure of the dispersion of scores around the mean; the most widely used measure of variability; there are 3 different kinds Why is std dev so popular? bc it is very reliable and provides info about the proportions w/in a distribution if you know the distribution's form What are the 3 different kinds of standard deviation? 𝞼, 𝑺, and ŝ 𝞼 standard deviation of the population; this is a parameter that measures a population's variability 𝑺 standard deviation of the sample; this is calculated with no interest in 𝞼... this is a statistic that measures a sample's variability ŝ standard deviation that estimates 𝞼; this is a statistic that estimates a population's variability deviation score formula vs. raw score formula use the deviation-score formula when you try to understand std dev, but raw score formula is much more efficient What is a deviation score? a raw score minus the mean of its distribution: X-X⎺, or X-μ When do I get a positive vs. negative deviation score? raw scores greater than the mean have positive deviation scores; raw scores that less than the mean have negative deviation scores; raw scores = to the mean have a deviation score of 0 What does the deviation score tell you? This value tells you the # of pts that a particular score deviates from, or differs from, the mean What is the deviation score formula? 𝑺=√[(Σ(X-X⎺)²)/N] or 𝞼= √[(Σ(X-μ)²)/N] Generally, how do you find the deviation score? Make a table in which the first column is X, the second column is the mean of each score, the 3rd column is the deviation score of X and the 4th column is the squared deviation score; input the values for each row of each column, then take the squared deviation scores and add them together; once added, divide this number by the number of scores present; then take the square root of that value for the final result What is the raw score formula? 𝑺 or 𝞼= √[((Σ𝑋²)-((Σ𝑋)²/N)/(N)] where: Σ𝑋²= sum of the squared scores (Σ𝑋)²=square of the sum of the raw scores N= # of scores How do you calculate ŝ? with the same formulas for 𝑺 or 𝞼, just with a denominator of "𝑁-1" ŝ formulas dev score formula: ŝ=√[(Σ(X-X⎺)²)/(N-1)] raw score formula: ŝ=√[((Σ𝑋²)-((Σ𝑋)²/N)/(N-1)] use 2nd formula for simple or grouped frequency distribution variance the square of the standard deviation what are the symbols for variance? 𝞼² & ŝ² formulas for variance same as the aforementioned formulas, just take away the square root symbol
chapter 4
What are the other descriptive statistics not described in central tendency or measures of variability? z-scores, outliers, boxplots, & effect size index What way can raw scores be turned into something meaningful? a raw score can be converted into a measure that signals its relationship to other scores with percentiles, z scores, and outliers z score a raw score expressed in standard deviation; it modifies an individual score's relationship to both the mean and std dev of its fellow scores; it basically allows you to see how well you do relative to others AdvertisementUpgrade to remove ads z score formula 𝓏= (𝑋 - 𝑋⎺)/𝑺 Where: (𝑋 - 𝑋⎺)=deviation score; & 𝑺= std dev Why must we divide the deviation score by the standard deviation to get the z score? the deviation score provides a raw score with no relation to the variability; dividing by the std dev allows for the variability of the distribution to be taken into account Why does dividing by std dev specifically allow for variability to be taken into account? Because it is a unit that measures variability What's a common name for the Z-score? standard score standard score a score expressed in standard deviation units; the z-score is one example of this which scores can be converted to z scores? any distribution of raw scores can be converted into a distribution of z scores; for each raw score, there is one z score positive vs. negative z-scores positive z scores represent raw scores greater than the mean negative z scores represent raw scores lesser than z score in terms of std dev the z score is the # of std dev's from the mean what do z scores compare? a single score's position on a distribution, or multiple scores' positions relative to each other as well as to the distribution If you grade on a curve, what are the z scores that determine grades A-F? +1.50=A +0.50 - +1.50=B -0.50 - +0.50=C -1.50 - -0.50D -1.50 & ↓ =F What type of statistics category do z scores fit into? it is used as both a descriptive statistic & inferential statistic z score as a descriptive statistic here, its range is limited to -3 - +3 or less z score as an inferential statistic here, values are not limited to + or - 3; instead, the value depends heavily on how different the 2 populations actually are outliers extreme scores How are outliers separated from non-extreme scores? they are separated from the others by 1.5xIQR or more beyond the 25th & 75th percentiles What is the most common way to identify outliers? w/ lower and upper outlier formulas Lower outlier anything beyond(to the left of): 25th percentile - (1.5xIQR) Upper outlier anything beyond(to the right of): 75th percentile + (1.5xIQR) boxplot a graph that shows a distribution's range, IQR, skew, median, mean, & outliers; it portrays all three characteristics of distributions (central tendency, variability, & form) in one graph how is median represented in a box plot? with a vertical line how is mean represented in a box plot? with a dot how is IQR represented in a box plot? with a box -lower edge=25th percentile -upper edge=75th percentile how is range represented in a box plot? with whiskers how is skew represented in a box plot? by the distance between mean & median symbols how are outliers represented in a box plot? with an asterix (*) effect size index this provides a measure of the magnitude of difference; symbol: 𝒹 effect size index formula 𝒹 = (μ₁-μ₂)/𝞼 where: 𝒹=Cohen's d... the index; μ₁=mean of 1st population; μ₂=mean of 2nd population (order of means doesn't matter); 𝞼-stdev of population How is 𝒹 measured? a small 𝒹=0.20 a medium 𝒹=0.50 a large 𝒹=0.80 This means that if the effect size index of men vs. women heights provided a 𝒹 of 0.25, then the height is not a huge difference between the two. Or if a drug provided an impact of 𝒹=0.89, it would have a large impact on the person receiving the drug
chapter 5
Probability For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. (proportion) Requirement for def of probability random samples, every individual has equal chance of being selected & probabilites must stay constant, so sampling w/ replacement Random Sample requires that each individual in the population has an equal chance of being selected AdvertisementUpgrade to remove ads Independent random sample requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next than one individual is selected Sampling with replacement To keep probabilities from changing from one selection to the next , it is necessary to return each individual to the population before you make the next selection unit normal table table used to identify the proportion of observations that lie at or beyond a particular z-score in a normal distribution percentile rank the percentage of individuals in the distribution with scores at or below the particular value, corresponds to proportion to left of score percentile when a score is identified by its percentile rank binomial distribution used when measurement classifies individuals into 2 categories p(A)=p and p(B)=q, pn and qn must both be at least 10 , like coin toss (H/T) or MC (correct vs wrong) or gender (M/F) mean of binomial distribution u=pn sd of binomial distribution o=sqr(npq) z score of binomial distribution x +/- 0.5 because normal curve is continuous and binomial is discrete, at least (2 or more) 2 hs in 10 tosses= 1.5, more than 15=15.5 Pr (A or B) Pr (A) + Pr (B) - (Pr A and B) - if not mutually exclusive Pr (ace and a jack) (ace then jack or jack then ace) so 2(4/52*4/52) 2 possible ways to occur w/o replacement 2(4/52+4/51) Pr(2 first years and 1 soph or 3 seniors) 3(200/700)^2 *(300/700) + (50/700)^3 Pr(1 head) 10(1/2)^1 * (1/2)^9
chapter 6
sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter distribution of sample means collection of sample means for all possible random samples of a particular size that can be obtained from a population. sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population AdvertisementUpgrade to remove ads central limit theorem As the size n of a simple random sample increases, the shape of the sampling distribution of x̄ tends toward being normally distributed. expected value of M The mean of the distribution of sample means is equal to the mean of the population of scores standard error of M the standard deviation of the distribution of sample means. The standard error provides a measure of how much distance is expected on average between a sample mean and the population mean. law of large numbers states that the larger the sample size, the more probable it is that the sample mean will be close to the population mean
chapter 7
