Psyc 10H Exam 1

Bimodal distribution

A bimodal distribution is a symmetrical distribution containing two distinct humps, each reflecting relatively high-frequency scores. At the center of each hump is one score that occurs more frequently than the surrounding scores, and technically the two center scores have the same frequency

Statistics versus parameters

A number that is the answer from a descriptive procedure that describes a sample of scores is called a statistic(Descriptive procedures). Different statistics describe different characteristics of sample data, and the symbols for them are letters from the English alphabet. However, a number that describes a characteristic of a population of scores is called a parameter(Inferential procedures). The symbols for different parameters are letters from the Greek alphabet.

What does "measure of central tendency" mean?

A score that summarizes the location of a distribution on a variable by indicating where the center of the distribution tends to be located based on: The first step in summarizing any set of data The scale of measurement used so that the summary makes sense given the nature of the scores. The shape of the frequency distribution the scores produce so that the measure accurately summarizes the distribution. descriptive statistics to envision the important aspects of the distribution

Simple frequency

A simple frequency distribution shows the number of times each score occurs in a set of data. The symbol for a score's simple frequency is simply . To find for a score, count how many times the score occurs. If three participants scored 6, then the frequency of 6 (its f ) is 3. Creating a simple frequency distribution involves counting the frequency of every score in the data. Start with a score column and an column. The score column has the highest score in the data at the top of the column. Below that are all possible whole-number scores in decreasing order, down to the lowest score that occurred. Thus, the highest score is 17 , the lowest score is 10 , and although no one obtained a score of 16, we still include it. Opposite each score in the column is the score's frequency: In this sample there is one 17, zero 16s, four 15s, and so on. There are 18 scores in the original sample, so N is 18. You can see this by adding together the frequencies in the f column graph of a frequency distribution shows the scores on the axis and their frequency on the axis.

Transformation

A systematic mathematical procedure for converting a set of scores into a different but equivalent set of scores transformations make scores easier to work with transformations make different kinds of scores comparable

What does the sum of the deviation scores equal?

Adding all of the negative deviations, we have the total distance that other scores are below the mean. If we add all of the positive and negative deviations together, we have what is called the sum of the deviations around the mean. The sum of all differences between the scores and the mean; symbolized as E(x-mean) total of the positive deviations will always equal the total of the negative deviations = equals zero

Participants

The individuals measured in a sample are called the participants (or sometimes, the subjects) and it is the scores from the sample(s) that constitute our data. As with a population, we will discuss a sample of scores as if we have already measured the participants in a particular situation.

What does "proportion of the total area under the curve" mean?

The proportion of the total area beneath the normal curve at certain scores, which represents the relative frequency of those scores. The proportion of the total area under the normal curve that is occupied by a group of scores corresponds to the combined relative frequency of those scores. he entire polygon occupies an area of 6 square inch . This total area corresponds to all scores. Say that the area under the curve between the scores of 30 and 35 covers 2 square inch . This area is due to the number of times these scores occur. Therefore, the scores between 30 and 35 occupy 2 out of the 6 sq inch created by all scores, so these scores constitute 2/6 , or .33 , of the sample. Thus, these scores occur .33 of the time in our sample so they have a relative frequency of .33. advantage of using the area under the curve is that we can get the answer without knowing or the simple frequencies of any scores The area under the normal curve corresponds to 100% of a sample, so a proportion of the curve will contain that proportion of the scores, which is their combined relative frequency.

Frequency distributions

The scores we initially measure in a study are called the raw scores The number of times a score occurs is the score's frequency, symbolized by the f. If we count the frequency of every score in the data, we create a frequency distribution. A distribution is the general name that researchers have for any organized set of data. answers our question about the different scores that occurred in our data and it does this in an organized manner.

Standard deviation (and the notation for the sample and the population standard deviation)

The square root of the sample variance or the square root of the average squared deviation of sample scores around the sample mean; symbolized by Sx if Sx is relatively large, we know that a large proportion of scores is relatively far from the mean, producing a wide distribution that is poorly summarized by the mean relatively small, we know the scores form a skinny distribution that is accurately summarized by the mean cannot be a negative number For any roughly normal distribution, the standard deviation should equal about one-sixth of the range ("average deviation" from the mean, the consistency in the scores, and how spread out the distribution is) sd pop: σx

What does it mean to have a relationship between two variables?

a pattern in which, as the scores on one variable change, the corresponding scores on the other variable change in a consistent manner. For example, say that we asked some students how long they studied for a test and their subsequent grades on the test. use the term association when talking about relationships A stronger relationship occurs the more that one group of similar Y scores is associated with one X score and a different group of similar Y scores is associated with the next X score.

Percent

a proportion multiplied by 100 (6/12)= .5(100)= 50%

Inferential statistics

are procedures for deciding whether sample data accurately represent a particular relationship in the population. Essentially, inferential procedures are for deciding whether to believe what the sample data indicate about the scores and relationship that would be found in the population are for deciding whether to believe what the sample data indicate about the scores and relationship that would be found in the population.

Descriptive statistics

are procedures for organizing and summarizing sample data so that we can communicate and describe their important characteristics. (When you see descriptive, think describe.) useful because they allow us to quickly and easily understand the data without having to look at every single score are used to summarize and describe the important characteristics of sample data and to predict an individual's Y score based on his or her X score.

What is the difference between the "N" and the "N-1" formula in calculating the standard deviation and variance?

biased: A formula for a sample's variability that involves dividing by N that is biased toward underestimating the corresponding population variability unbiased:A formula for a sample's variability that involves dividing by N-1 that equally often under and over-estimates the corresponding population variability the problem with the biased estimators (Sx and S^2x) is that these formulas divide by N. Because we divide by too large a number, the answer tends to be too small. Instead, we should divide by N-1. This is a slightly smaller number that produces a slightly larger answer, which is a more accurate estimate of the population's variability. By dividing by N-1 we create new formulas called the unbiased estimators of the population variance and standard deviation estimated population standard deviation-The unbiased estimate of the population standard deviation, calculated from sample data using degrees of freedom N-1; symbolized by sx estimated population variance-The unbiased estimate of the population variance, calculated from sample data using degrees of freedom N-1 ; symbolized by s^2x sx s^2x as the inferential variance and the inferential standard deviation, because the only time you use them is to infer the variance or standard deviation of the population based on a sample

Continuous versus discrete scales

continous: allows for fractional amounts; it "continues" between the whole-number amounts, so decimals make sense. The variable of age is continuous because someone can be 19.3762 yrs old. discrete: which are measured only in whole amounts. Here, decimals do not make sense. For example, whether you are male or female or in first or second grade are discrete variables because you can be in one group or the other, but not in-between.-shoe size Usually researchers assume that nominal or ordinal variables are discrete and that interval or ratio variables are at least theoretically continuous.

Proportion

decimal number between 0 and 1 that indicates a fraction of the total. To transform a number to a proportion, simply divide the number by the total.

What does "measures of variability mean?"

describe the extent to which scores in a distribution differ from each other. With many large differences among the scores, our statistic will be a larger number, and we say the data are more variable or show greater variability. (Measures of variability communicate how much the scores differ from each other, which in turn determines how spread out the distribution is and how accurately they are summarized by the mean.)

Range

describe variability is to determine how far the lowest score is from the highest score. The descriptive statistic that indicates the distance between the two most extreme scores in a distribution involves only the two most extreme scores, so it is based on the least typical and often least frequent scores. Therefore, we usually use the range as our sole measure of variability only with nominal or ordinal data.

Frequency polygon

don't create a histogram when we have a large number of different interval or ratio scores but when a histogram is unworkable, we create a frequency polygon. Construct a frequency polygon by placing a data point over each score on the axis at a height corresponding to the appropriate frequency. Then connect the data points using straight lines. Because each line continues between two adjacent data points, we communicate that our measurements continue between the two scores on the X axis and therefore that this is a continuous variable (height=frequency)

Data point

dot plotted on a graph to represent a pair of x and y scores.

Sample

is a relatively small subset of a population that is intended to represent, or stand in for, the population. Thus, we might study the students in your statistics class as a sample representing the population of all college students enrolled in statistics. (scores from one student can be a sample representing the population of all scores that the student might produce. Thus, a population is any complete group of scores that would be found in a particular situation, and a sample is a subset of those scores that we actually measure in that situation.)

Population

is the entire group of individuals to which a law of nature applies. The population might be broadly defined (such as all animals or all humans), but it can be more narrowly defined (such as all women, all four-year-old English-speaking children in Canada, or all presidents of the United States) (population of scores)

Relative frequency

is the proportion of N that is made up by a score's simple frequency. Recall that a proportion indicates a fraction of the total, so relative frequency indicates the fraction of the entire sample that is made up by the times that a score occurs. Thus, whereas simple frequency is the number of times a score occurs, relative frequency is the proportion of time the score occurs. The symbol for relative frequency is rel f .(f/N) To transform relative frequency into simple frequency, multiply the relative frequency times

Normal distribution

mathematical properties define this polygon, in general it is a bell-shaped curve. Call it a normal curve or a normal distribution or say that the scores are normally distributed.represents an ideal population. produces so many different scores. reflects an infinite number of scores. might occur- never reach—a frequency of zero, so the curve approaches but never actually touches the axis. highest frequency is the middle score.symmetrical, meaning that the left half below the middle score is a mirror image of the right half above the middle score. farther a score is from the center of the distribution, the less frequently the score occurs.

What value would you use to predict an individuals' score?

mean is our best prediction about the score that any individual obtains. Because it is the central, typical score, we act as if all the scores were the mean score, and so we predict that score every time When we use the mean to predict scores, a deviation (x-mean) indicates our error: the difference between the mean we predict for someone and the X that he or she actually gets. If we predict any score other than the mean, the total error will be greater than zero. A total error of zero means that, over the long run, we overestimate by the same amount that we underestimate.

Mode

most frequently occurring score normal curve, with the highest point at the mode-unimodal-one score qualifies as the mode (one score qualifies as the mode) mode is the most frequently occurring score in the data and is usually used to summarize nominal scores Compute the mode with nominal data or with a distinctly bimodal distribution of any type of scores.

Measurement scales: nominal, ordinal, interval, and ratio

nominal:each score does not actually indicate an amount; rather, it is used for identification. (When you see nominal, think name.) License plate numbers and the numbers on football uniforms reflect a nominal scale. The key here is that nominal scores indicate only that one individual is qualitatively different from another, so in research, nominal scores classify, or categorize, individuals ordinal: Here the scores indicate rank order: anything that is akin to 1st,2nd ,3rd ... is ordinal. (Ordinal sounds like ordered.) In our studying example, we'd have an ordinal scale if we assigned a 2 to students who scored best on the test, a to those in second place, and so on. The key here is that ordinal scores indicate only a relative amount—identifying who scored relatively high or low. Also, there is no zero in ranks, and the same amount does not separate every pair of adjacent scores: 1st may be only slightly ahead of 2nd, but 2nd may be miles ahead of 3rd. interval: Here each score indicates an actual quantity, and an equal amount separates any adjacent scores. (For interval scores, remember equal intervals between them.) However, although interval scales do include the number , it is not a true zero it is incorrect to make "ratio" statements that compare one score to another score. ratio: do the scores reflect the true amount of the variable that is present. Here the scores measure an actual amount, there is an equal unit of measurement, and 0 truly means that none of the variable is present. The key here is that you cannot have negative numbers because you cannot have less than nothing. Also, only with ratio scores can we make "ratio" statements, such as 4" is twice as much as 2."

PEMDAS

reading a formula, remember there is an order of precedence of mathematical operations. You may have learned this as PEMDAS or maybe as " Please Excuse My Dear Aunt Sally. " first compute inside any Parentheses, then compute Exponents (squaring and square roots), then Multiply or Divide, and finally, Add or Subtract.

Statistical notation: for samples and populations

refers to the standardized code for symbolizing the mathematical operations performed in the formulas and for symbolizing the answers we obtain sample: x pop: μ

What does it mean that a score is in the "tail" of a distribution

scores that are relatively far above and below the middle score of the distribution are called the "extreme" scores. Then, the far left and right portions of a normal curve containing the low-frequency, extreme scores are called the tails of the distribution

Median

simply another name for the score at the 50th percentile. Recall that the 50th percentile is the score having 50% of the scores at or below it. Thus, if the median is 10 , then 50% of the scores in the sample are either at or below 10 only one score can be the median and the median will usually be around where most of the scores in the distribution are located summarize ordinal or highly skewed interval or ratio scores. reflects only the frequency of scores in the lower 50% of the distribution, not for describing the central tendency of normally distributed interval or ratio scores median to summarize a skewed distribution Compute the median with ordinal scores or with a skewed distribution of interval or ratio scores.

Negative versus positively skewed distributions

skewed distribution is similar to a normal distribution except that it has only one pronounced tail Positive: contains extreme high scores that have low frequency but does not contain low-frequency, extreme low scores. the tail slopes away from zero, toward where the higher, positive scores are located. negative: contains extreme low scores that have a low frequency but does not contain low-frequency, extreme high scores.the pronounced tail is over the lower scores, sloping toward zero, toward where the negative scores would be

What does it mean that "we always apply descriptive statistics to scores from the dependent variable?

summarizing an experiment, we first have specific descriptive procedures for summarizing the scores in each condition and for describing the relationship. For example, it is simpler if we know the average error score for each study time. Notice, however, that we apply descriptive statistics only to the dependent scores. Above, we do not know what error score will be produced in each condition, so errors is our "I Wonder" variable that we need help making sense of. We do not compute anything about the conditions of the independent variable because we created and controlled them. (Above, we have no reason to average together 1 ,2 ,3 , and 4 hours.) Rather, the conditions simply create separate groups of dependent scores that we examine. Then the goal is to infer that we'd see a similar relationship if we tested the entire population in the experiment, and so we have specific inferential procedures for experiments to help us make this claim.

Dichotomous variable

when a discrete variable has only two possible categories or scores, it is called a dichotomous variable. Male/female or living/dead are dichotomous variables.

Bar graph versus histogram

Bar: A frequency distribution of nominal or ordinal scores is graphed by creating a bar graph. In a bar graph, a vertical bar is centered over each score on the axis, and adjacent bars do not touch. The reason we create bar graphs with nominal and ordinal scales is because researchers assume that both are discrete scales: You can be in one group or the next, but not in-between. The space between the bars in a bar graph indicates this. Histogram: We create a histogram when plotting a frequency distribution containing a small number of different interval or ratio scores. A histogram is similar to a bar graph except that in a histogram adjacent bars touch. interval and ratio scales are assumed to be continuous: They allow fractional amounts that continue between the whole numbers. To communicate this, these scales are graphed using continuous (connected) figure

Variance (and the notation for the sample and the population variance)

By finding the average squared deviation, we compute the variance the average of the squared deviations of scores around the sample mean capital S = sample = Sx^2 bad: squaring the deviations makes them very large, so the variance is unrealistically large variance is rather bizarre because it measures in squared units variance does communicate the relative variability of scores pop variance: σ²x pop: Both are ways of measuring how much the scores are spread out in the population. The variance is the average squared deviation of scores around μ, and the standard deviation is somewhat like the "average" amount the scores differ from μ.

Cumulative frequency

Cumulative frequency is the frequency of all scores at or below a particular score. The symbol for cumulative frequency is cf. To compute a score's cumulative frequency, we add the simple frequencies for all scores below the score to the frequency for the score, to get the frequency of scores at or below the score. Begin with the lowest score. each time adding the frequency for a score to the cumulative frequency for the score immediately below it. cf for the highest score equals N

Experiment: condition, levels, treatment

In an experiment the researcher actively changes, or manipulates, one variable and then measures participants' scores on another variable to see if a relationship is produced. The purpose of an experiment is to produce a relationship in which, as we change the conditions of the independent variable, scores on the dependent variable tend to change in a consistent fashion. To see the relationship and organize your data, diagram an experiment Condition:selects the conditions of the independent variable. A condition is the name for a specific amount or category of the independent variable that creates the specific situation under which participants are examined. Level/treatmentA condition is also known as a level or a treatment: By having participants study for 1 hour, we determine the specific "level" of studying that is present, and this is one way we "treat" the participants. level:In ANOVA, each condition of the factor (independent variable); also called treatment treatment:The conditions of the independent variable; also called levels

Variable

In research the factors we measure that influence behaviors—as well as the behaviors themselves—are called variables. A variable is anything that, when measured, can produce two or more different scores. A few of the variables found in behavioral research include characteristics of an individual, like your age, race, gender, or intelligence or your personality type, political affiliation, or physical attributes. Variables also measure reactions, such as how anxious, angry, or aggressive you are or how attractive you find someone. If a score indicates the amount of a variable that is present, the variable is a quantitative variable. A person's height cannot be measured in amounts, but instead, a score classifies an individual on the basis of some characteristic(gender)

Independent and dependent variables

Independent: refers to the variable that is changed or manipulated by the experimenter. Implicitly, it is the variable that we think causes a change in the other variable. In our studying experiment, we manipulate study time because we think that longer studying causes fewer errors. Thus, amount of study time is our independent variable. (if we want to examine whether gender is related to some behavior, we would select a sample of females and a sample of males. In our discussions, we will call such variables independent variables because the experimenter controls them by controlling a characteristic of the samples (quasi-independent variables)) Dependent: measures a participant's behavior under each condition. We measure this behavior because we expect it will be influenced by the particular condition of the independent variable that is present. Thus, a participant's high or low score on this variable is supposedly caused or influenced by—depends on—the condition that is present.

Deviation score

The distance separating a score from the mean indicating the amount the score "deviates" from the mean. A score's deviation is equal to the score minus the mean, or in symbols: (raw score - mean) The number, which indicates distance from the mean (which is always positive), and the sign, which indicates direction from the mean

What is your best guess for the mean of a population?

Usually we have interval or ratio scores that form at least an approximately normal distribution, so we usually describe the population using the mean. mean of a population is μ the Greek letter (pronounced "mew") estimate μ based on the mean of a random sample. If, for example, a sample's mean in a particular situation is 99 , then, assuming the sample accurately represents the population, we would estimate that μ in this situation is also 99. We make such an inference because it is a population with a mean of 99 that is most likely to produce a sample with a mean of 99

Relationship between measures of variability and "consistency" of scores, and between the accuracy at which we can use a single score to describe individual scores within the distribution

the opposite of variability is consistency. Small variability indicates few and/or small differences among the scores, so the scores are consistently close to each other (indicating that similar behaviors are occurring) how accurately the distribution is described by our measure of central tendency. Our focus will be on the mean and normal distributions: The greater the variability, the less accurately the mean will summarize the scores. Conversely, the smaller the variability, the closer the scores are to each other and to the mean. difference between two scores can be thought of as the distance that separates them. larger differences indicate larger distances between the scores, so measures of variability indicate how spread out a distribution is

Mean (and the notation for the sample and population mean)

the score located at the mathematical center of a distribution x bar = the sample mean do not use the mean when describing nominal or ordinal data = need average Always compute the mean to summarize a normal or approximately normal distribution: The mean is the mathematical center of any distribution, and in a normal distribution, most of the scores are located near this central point, and they are balanced around it. mean will inaccurately describe a skewed (nonsymmetrical) distribution mean is always pulled toward the tail of any skewed distribution mean is pulled toward the extreme tail and is not where most scores are located pop: mew samp: x bar

Percentile in small and large samples

transform cumulative frequency into a percent of the total. A score's percentile is the percent of all scores in the data that are at or below the score. Thus, for example, if the score of 80 is at the 75th percentile, this means that 75% of the sample scored at or below 80. first divide the score's cf by N , which transforms the cf into a proportion of the total sample. Then we multiply this times 100 , Percentile describes the scores that are lower than a particular score, and on the normal curve, lower scores are to the left of a particular score. Therefore, the percentile for a given score corresponds to the percent of the total area under the curve that is to the left of the score. Then percentile becomes the percent of scores below a particular score. This is acceptable if we are describing a large sample or a population because those participants at the score are a negligible portion of the total small sample, we should not ignore those at the score, because those participants may actually constitute a sizable portion of the sample. small samples, percentile is calculated and defined as the percent of scores at or below a particular score

Correlational study

we simply measure participants' scores on two variables and then determine whether a relationship is present. Unlike in an experiment in which the researcher actively attempts to make a relationship happen, in a correlational design the researcher is a passive observer who looks to see if a relationship exists between the two variables. As usual, we want to first describe and understand the relationship that we've observed in the sample, and correlational designs have their own descriptive statistics for doing this. Then, to describe the relationship that would be found in the population, we have specific correlational inferential procedures.