Week 3: Measurement Scales and Descriptive Statistics

What is the most meaningful: mean, median or mode for data on the nominal scale?

Only the mode is meaningful

True/false: The categories are mutually exclusive in a nominal scale and it is impossible to assign a single observation in two different categories at once.

True

True/false: We can add, subtract, multiply, and divide ratio data

True

What is Central tendency? What are the measures of it?

-Numerical indices that describe the "typical" nature of the data and to reflect different concepts of the "center" of the distribution. -Is a single value that attempts to describe a set of data by identifying the central position within that set of data -Mean, median, mode

What is an example of Ratio data?

Height, weight, distance, time have true zero points(not possible to have a negative on either of these, all of these have true zero points)

What is a Histogram?

-A bar graph, composed of a series of columns, each representing one score or class interval. -The measured variable is on the X-axis and the frequency of that score is shown on the Y-axis.

What are Measures of Variability? What is variance?

-Characterize the differences that exist among the scores as well as the central tendency of the data. -Variance means people are more spread out and easier to distinguish variability. -Range, Variance, Standard Deviation, Coefficient of variation(not going to cover), Percentile -Measures of variability reflect the extent to which scores differ among themselves. -Restated, measures of variability answer the question, "how much spread is there in test scores?"

True/false: The median is much more affected by extreme scores than is the mean.

-False -The mean is much more affected by extreme scores than is the median.

Nominal Scales cont.

-The numbers in a nominal scale have no meaning by themselves; they are not ordered in any way e.g., numbers on football player numbers just identify the players)(Couldn't add the numbers of linemen and have it means something) -Placement of an observation in a particular category (e.g., 1 = females) simply indicates that observation is different from observations in other categories (e.g., 2 = males). The numbers do not imply that one number is more or less different. Doesn't mean that 2 is greater than 1.

What is Normal Distribution? What is another name for it?

-Very important -An important statistical concept because so many biological, psychological, and social phenomena manifest themselves in populations according to this shape -Normal distribution = normal curve = bell curve = Gaussian distribution

Describe the chart and how the sum of squares affects it (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

1. Square each deviation score. Ex: -18.63 times -18.63=347.08 2. Do that for the rest of the deviations 3. Add all of the deviations to get 1044.63.

What is an example of measurement?

A certain degree of spinal curvature be present to indicate a diagnosis of scoliosis OR the functional status of an elderly patient to determine the levels of assistance that will be required when the patient returns home

What is an example of construct?

The chart assesses physical functioning. Physical functioning is an abstract variable since the entirety of physical functioning cannot be seen at one time but it can be inferred by measuring relevant or correlated behavior. Can measure how a person with different levels of physical functioning could behave, look or feel in certain circumstances. 1. Are you able to get in and out of bed? That item has a relationship with the construct of physical functioning. Physical functioning itself is an abstract thing that we all understand but hard to observe in its entirety directly. Have to observe behaviors that relate to the construct in order to assess it.

True/false: Nominal and ordinal data are not expected to be normally distributed.

True

What happens if there is no spread in the scores?

When there is no spread in the scores, the range, variance, and standard deviation will equal 0.

What is it called when we express scores in terms of standard deviation units?

When we express scores in terms of standard deviation units, we are using standardized scores, also called Z-scores.

How do you calculate Z Scores?

where: X = a particular person's score M = the mean score SD = the standard deviation

Comparing measures of Central tendency

All 3 measures of central tendency can be applied to variables on the interval or ratio scale, although the mean is most useful.

What is the most meaningful: mean, median or mode for ordinal data?

Both the median and mode can be applied

Two types of statistics

Descriptive Statistics Inferential Statistics

True/false: We cannot add, subtract, multiple, and divide in an Interval Scale.

We CAN add, subtract, multiple, and divide...but these operations cannot be used to interpret actual quantities because the zero point is arbitrary.

Descriptive Statistics

...

What are the percentiles for the proportions of the normal curve?

-34% is between the mean and ± 1SD. -68% of scores fall between ± 1SD below the mean and 1SD above the mean. -95% fall between ±2 SD of the mean. -99.7% between ±3SD of the mean. -Because we can never discount extreme values at either end, we never account for the full 100% -This information can be used as a bases for interpreting SDs. For example, if we are given -M=65 ± 6.06, we can estimate that approximately 68% of individuals in the sample have scores between 58.94 and 71.06

What is a useful way to demonstrate visually the spread of scores in a distribution?

-A box plot graph, also called a box-and-whisker plot, is a useful way to demonstrate visually the spread of scores in a distribution. -"whiskers" may represent highest and lowest scores or may represent the the 10th and 90th %iles. In this case, outliers beyond those values are shown as circles.

What is Standard Deviation? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-A limitation of the variance is that it was calculated using the squares of the deviation scores. -So to bring the number back into the original units, we take the positive square root of the variance. This is called the standard deviation. -Represents the average degree to which people differ from one another -The SD is usually reported along with the mean that the data are characterized according to both central tendency and variability. -May be expressed as: mean±SD -Ex: 83.63±12.22 (mean+SD) -X=test scores -X(with a line on top)=mean of test scores -N= the number of participants

What are Standardized Scores?

-Also called z-scores -The number of standard deviations that a given value is above or below the mean of the distribution -Ex: A score of 58 can be expressed as a z-score of -1.0, the minus sign indicating that it is one SD unit below the mean A score of 88 is transformed to a z-score of +2.0 or two standard deviations above the mean.

Define an Interval Scale

-An interval scale possesses the rank-order characteristics of an ordinal scale, but also demonstrates known and equal distances or intervals between the units of measurement. -Therefore, relative difference and equivalence within a scale can be determined. -What is not supplied by an interval scale is the absolute magnitude of an attribute because interval measures are not related to a true zero. -This leads to another issue with zeros in the interval scale: Zero doesn't mean that something doesn't exist. For example, the year 0 doesn't imply that time didn't exist. And similarly, a temperature of zero doesn't mean that temperature doesn't exist at that point. Arbitrary zeros (and the inability to calculate ratios because of it) are one reason why the ratio scale — which does have meaningful zeros — is sometimes preferred.

What is a "floor effect?" What is an example of it?

-C is an example of a floor effect: when a lot of people cluster on the low end. This means that if you give a measure and everyone scores the lowest possible thing, it means that they might have a level of the trait that is lower than the scale permits to measure. -Ex: Giving preschool children an IQ test designed for adults would likely show many of the test-takers with scores near the lowest standard score for adult test-takers. Aquiles example: Have a 5 year old and given 3 words: obloquy, persistent, treacherous. They will most likely not get these words and therefore show a floor effect(positively skewed).

What are Percentiles?

-Describe a score's position within a distribution. -Divide the data into 100 equal portions -A particular core is located in one of these positions, which represents its position relative to all other scores. -The percentage of scores AT or BELOW the target score -Ex: If a student took a college entrance exam and scored in the 92nd percentile, that individual's score was higher than 92% of those that took the test.

What are inferential statistics?

-Makes inferences and predictions about a population based on a sample of data taken from the population in question -Helps researchers make decisions about what their data mean and what inferences they can make from them. (Ex: Is this group older than that group?)

What are the functions of measurement?

-Describe the quality(gender or geographic location) or quantity(age or blood pressure) of a variable -Make absolute decisions based on a criterion --Is your vision good enough to fly a plane? -Can help choose between two courses of action -Assess change --Is someone improving with treatment?(Helpful to be able to put numbers behind it. If patient, partner and you think they're changing but the numbers aren't showing changing maybe you have a bad measurement. Doesn't mean you shouldn't measure but instead develop better measurement). -Original purpose of SAT scores was to predict freshman year grades. Does it?

What is the formula for Sample variance? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-Deviation scores are obtained using X(with a bar on top), not u. -Because sample data do not include all the observations in a population, the sample mean is only an estimate of the population mean. To compensate for this bias, the SS is divided by (n-1) to calculate the sample variance, given the symbols s2(squared).

Levels of Measurement: Nominal, Ordinal, Interval, Ratio

-Each level of measurement builds on the other. -For instance, ordinal data contains all the information of nominal data, plus some other things. -Likewise, interval data contains all the information of nominal and ordinal data, plus some other things.

What are Error Bars?

-Error bars MAY represent the SD -1 SD -2 SD -Standard error of the mean (SEM)(later lecture) -ALWAYS READ THE CAPTION to know what the error bar is telling you

What is an example of Ordinal Data?

-Ex: Amoeba, ant, horse, dinosaur. Difference between them is not the same (between ameba and ant vs. horse and dinosaur). -The intervals between ranks may not be consistent and may not be known

What are some examples of Standardized Scores?

-Ex: If a patient has a pules of 58 beats/min, the implication of that value is evident only if we know where that score falls in relations to a distribution of normal pulse rates. If we know that the average is 68 and the s(SD)=10 for a given sample, then we know that an individual score of 58 is one SD below the mean. -Ex: I made a new test of language competencies and your child produced a score of 80." (is this out of 100 then not so good or out of 50 then it's great).

True/false: The median represents the 75th percentile.

-False -The median represents the 50th percentile.

True/false: We can add, subtract, multiply, or divide data on the nominal level.

-False -We cannot add, subtract, multiply, or divide data on the nominal level. (Numbers indicate just what group they're under)

Define Ratio data

-Highest level of measurement with an actual 0 point which represents the total absence of whatever property is being measured. -An interval scale with an absolute zero point that has empirical, rather than arbitrary, meaning. -A score of zero represents a total absence of whatever property is being measured -Negative values are not possible(because there is an absolute zero)

What is construct?

-Invented names for abstract variables of a person that often cannot be measured directly, but can be assessed by measuring relevant or correlated behaviors that are observable -Measurement of a construct is based on expectations of how a person who possesses the specified trait would behave, look, or feel in certain situations.

What are proportions of the normal curve? Where are most score clustered around?

-Is smooth, symmetrical and bell-shaped, with most of the scores clustered around the mean -Vertical axis of the curve represents the frequency of scores, which decreases steadily as scores move in a negative or positive direction away from the mean. -Because of standard properties, we can determine the proportional areas under the curve represented by the standard deviations in a normal distribution

Shape of a Distribution: Kurtosis: What are the different kinds of shapes?

-Kurtosis: How high up or how sharp the peak of the distribution will go -Normal distribution(green): Mesokurtic -Flat top distribution(blue): Platykurtic -Abnormally spiky(red)- Leptokurtik

What is the Median?

-Line up scores lowest to highest first. the value at the middlemost score of a distribution of scores-the score that divides a frequency distribution into halves. -Odd number of scores = the median is the middle number -Even number of scores = the median is the midpoint between the two middle scores

What is a Nominal Scale?

-Lowest level of measurement -A good way to remember this is that "nominal" sounds a lot like "name" and nominal scales are kind of like "names" or labels. -Nominal scales are used for labeling variables without any quantitative value. Could be simply called "labels." -Objects or people are assigned to categories according to some criterion. -Categories may be coded by name, number, letter or symbol, although none of these have any numerical significance (they are used purely as labels for identification.

What is the formula to calculate the mean?

-M= the mean of the sample -The Sigma (E)= "the sum of" -X= the amount of scores in the group -Lowercase n=the number of participant taking the test

What is measurement?

-Process of assigning numerals to variables to represent quantities of characteristics according to certain rules. -Measurement provides a mechanism for achieving a degree of precision in this understanding, so that we can describe physical or behavioral characteristics according to their quantity, degree, capacity, or quality -Used to get a degree of precision

What are Quartiles?

-Quartiles divide a distribution into four equal parts, or quarters So there are three quartile dividers (Q1, Q2, Q3) -The score at the 50th %ile or Q2 is the median. -The distance between the first and third quartiles , Q1-Q3, is called the interquartile range, which represents the middle 50% of the distribution. -Ex: box-and-whisker plot

What is an example of Bimodal?

-Reviews for anything, will show love or hate something. -Mostly bimodal with 5's and 0's.

Graphing Frequency Distributions

-Stem-and-leaf plot -Historgram

What is an example of a deviation score? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-Take sum of scores and get mean=83.62. -Subtract that mean from each score (65-83.63=-18.63) to get the deviation score -However, the first three deviation scores are negative values because these scores are smaller than the mean. -Have to square the deviation score to get rid of the minus signs

What are Z Scores?

-The Z score is the basic standardized score. It can be transformed into other standardized scores -Z scores provide information about the relative position of a score by describing it's distance from the mean. -Unlike the variance and SD, which are calculated for a group of participants, Z-scores can also be used to provide information regarding how far a particular individual's score is above or below the mean.

What is the mean? What is it abbreviated by?

-The most commonly used measure of central tendency as well as the most precise measure of it. -Found by adding all the scores in a batch and then dividing by the number of scores -Abbreviated with the Greek letter, mu

What is the Range and how do you calculate it?

-The range is simply the distance between the highest and the lowest score in a distribution. -Range = the highest score - lowest score. Ex: Range = 40 - 10 = 30 -Because the range depends solely on the two most extreme scores in a distribution, it is very untrustworthy and/or unreliable measure of variability. -It doesn't't say anything about the scores between the two extremes -A shift in a single extreme score may greatly affect the range(highest or lowest number). -All things being equal, as the number of participants increases, the range will tend to increase - and this is another problem with the range.

Define Ordinal Data

-The term rank is often used to signify the placement of objects or events on an ordinal scale. -An ordinal scale is distinguished from a nominal scale by the additional property of order among the categories included on the scale. -Data are organized into adjacent categories exhibiting a "greater than - less than" relationship -With this category of measurement, we can talk about the order of a set of objects, but we cannot say how much larger one thing is than another.

What is Sum of Squares (SS)? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-This is when you square each deviation score to get rid of the minus signs -The sum of the squared deviation scores is called the sum of squares (SS). -As variability increases, the sum of squares will be larger.

What can be calculated using an Interval Scale?

-Thus, we can calculate a M(mean) and SD(standard deviation) with interval data. -As a consequence of being measured with equal units, and being able to calculate an M and SD, it is possible for interval data to be normal

True/false: Extreme scores cause the score distribution to become either positively skewed or negatively skewed.

-True -As a result, although the mean is a more precise statistic than the median, it is less stable. When the score distribution has extreme scores and becomes either positively or negatively skewed, the median is the better measure of central tendency.

True/false: Adding more scores of the same construct won't change the mean as much as other indices of central tendency.

-True -However, the mean is more affected by extreme scores, outliers, (i.e., a few scores in the distribution that are very high or very low will affect it dramatically).

True/false: Point is to get neither a positive or negatively skewed distribution.

-True -Looking to get a normal distribution

True/false: The mean, median and mode have the same value for a normal distribution

-True -Most scores are clustered around mean: The mean, median, and mode have the same value.

True/false: The best way to determine central tendency on a set of ordinal data is to use the mode or median

-True -The best way to determine central tendency on a set of ordinal data is to use the mode or median; the mean cannot be defined from an ordinal set -Ex: cannot get mean of an amoeba, ant, horse and dinosaur since you can't add and then divide those

What is a Stem-and-leaf plot?

-Useful for presenting the pattern of distribution of a continuous variable -Leaf = the last or rightmost single digit of each score -Stem = the remaining leftmost digits -Ex: Leaf is the right most digit score. Ex: Stem 6 and leaf is 08 so scores you get are a 60 and a 68.

What are descriptive statistics?

-Uses the data to provide descriptions of the population, either through numerical calculations or graphs or tables. -A set of statistics used to characterize the shape, central tendency, and variability within a set of data, often with the intent to describe a population. -Ex: Of this group of people, on average, what was their age?

What is Variance? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-Variance (s2) -The SS is is limited because it is influenced by the sample size. That is, as n increases, the SS will also increase simply because there are more scores. -Therefore, to correct for this the variance is computed, which corrects for the sample size. -Two different equations to compute: Population variance or Sample Variance

What is the formula for Population variance? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-When the population is known, deviation scores are obtained by X-u (almost never true in CSCD research) -The variance is symbolized by a o2.

What is Frequency distribution?

-a table of rank ordered scores that shows the number of times each value occurred (i.e., its frequency) -how many times that score occurred in a distribution -Percent: what percentage of that score is of the data

What is the Deviation Score? (DON"T NEED TO MEMORIZE HOW TO CALCULATE)(JUST KNOW WHAT IT IS)

-distance that each score falls from the mean -samples with larger deviation scores will be more variable around the mean.

What is cumulative percentage?

-the percentage of scores that fall at or below the target score. -when you add the 11, 10, 9 and their percentages, you can move over on the chart and see the cumulative percent will be 12.5 which is the percent's added.

What is the Mode? What kind of different modes are there?

-the score(s) that occur most frequently in a distribution. -the value of the most common score. Look at a histogram and find the peak. -Unimodal (one mode) -Bimodal (two modes) -Multimodal (more than 2 modes) -When a distribution has more than one mode, neither the mean or median accurately describes the distribution. -In fact, bimodal distributions cannot be described by a single value such as the mean or median (Sprinthall, 2007).

What is Distribution and what is it marked by?

-the total set of scores for a particular variable -The total number of scores in the distribution is given the symbol n. -Ex: n=48 (sometimes uppercase N is assigned to true populations, whereas lowercase n is indicating a sample, will almost always use the lowercase n)

Shapes of Distributions: What are the different types and what do they mean?

A- uniform distribution (ever score was produces equally). B- Normal Distribution C-positively skewed distribution(means they are clustered to lower end and fewer high scores. Ex: incomes in US, most people have incomes in the lower and middle end will be clustered here). If tail is pointing toward a positive side, it is a positive skew. Floor effect. D- negatively skewed distribution: ceiling effect (if tail is pointing toward a negative end it is negatively skewed).

What is a "ceiling effect?" What is an example of it?

D is an example of a ceiling effect: Ex: If you develop a measure of physical functioning for the average person. One item assessed was "how well can you run a 10k." If assess military service members, they could all do it with no training and would score at the top of the scale because they could all do it. Therefore not a good measurement scale since they will all be closer to the top. Aquiles example: Have a 5 year old and given 3 words: ball, red, car. They will most likely get these words and therefore show a ceiling effect(negatively skewed)

What is an example of an Interval Scale?

Ex: temp measurement. Difference between 60 degrees and 70 degrees are the same as the difference between 30 degrees and 40 degrees.

What is a good way to remember the shapes of distributions?

To help remember the different Leptokurtic=think of two kangaroos standing with heads toward each other (leap)Platykuetic=platypus

True/false: We cannot add, subtract, multiply, or divide data on the ordinal level.

True