PSY230 Midterm

Know the z-score distribution will have exactly the same shape as the original distribution of scores ,will always have a mean of zero and will always have a standard deviation of 1.

-Z-Score Distribution will have exactly same shape as original Distribution -Mean=0 -Standard deviation of 1

Understand the relationship between-population, sample, parameter, and statistic. Identify examples for each.

A parameter is a number describing a whole population, while a statistic is a number describing a sample Parameter- population mean population- the entire group that you want to draw conclusions about statistic- sample mean sample- the specific group that you will collect data from -For example, say you want to know the mean income of the subscribers to a particular magazine—a parameter of a population. You draw a random sample of 100 subscribers and determine that their mean income is $27,500 (a statistic).

Describe the effect on the mean and calculate the outcome for each of the following: changing a score, adding or removing a score, adding or subtracting a constant from each score, and multiplying or dividing each score by a constant.

Adding/Removing a Score If we add a data point that's above the mean, or take away a data point that's below the mean, then the mean will increase. If take away a data point that's above the mean, or add a data point that's below the mean, the mean will decrease. My mean is 7 If I add 6 to each score: my new mean is 13, because you just have to add 6 to the original mean, same as if I subtracted 6 to each score my mean would've been 1 If I multiply each score by 2: my new mean is 14, because you just have to multiply your original mean by 2, so if I was dividing by two it would be 3.5

Know the effect on standard deviation and calculate the outcome for each of the following: adding or subtracting constant from each score, and multiplying or dividing each score by a constant.

Adding/subtracting a constant to each value in a data set does not change the distance between values so the standard deviation remains the same. multiplying/dividing by a constant means that the standard deviation will be multiplied or divided by the same constant.

Know the definition of biased and unbiased statistics. (the sample mean and the sample variance are unbiased statistics).

Biased statistics- if it is calculated in such a way that it is systematically different from the population parameter being estimated. Unbiased statistics- an accurate statistic that's used to approximate a population parameter. find the errors by subtracting each estimate from the actual or observed value. Add up all the errors and divide by the number of estimates to get the bias. If the errors add up to zero, the estimates were unbiased, and the method delivers unbiased results.

Calculate the variance and the standard deviation for a sample.

Calculate the sample standard deviation of their responses: 2,2,5,7 Step 1: Find the mean. 2+2+5+7= 16/4 = 4 The sample mean is 4. Step 2: Subtract the mean from each score. 2-4= -2 2-4= -2 5-4= 1 7-4= 3 Step 3: Square each deviation. -2^2= 4 -2^2= 4 1^2= 1 3^2= 9 Step 4: Add the squared deviations. 4+4+1+9= 18 Step 5: Divide the sum by one less than the number of data points. 18/ 4-1 = 18/3 = 6 Take the square root of the result from Step 5. square root of 6 ​≈ 2.45 The sample standard deviation is approximately 2.45

Calculate the variance and standard deviation for the population.

Calculate the standard deviation of their scores: 6, 2, 3, 1 Step 1: Find the mean. μ=6+2+3+1/4 = 12/4 = 3 The mean is 3 points. Step 2: Subtract the mean from each score. Score: xi, Deviation: (xi−μ) 6−3=3 2-3=-1 3-3=0 1-3=-2 Step 3: 3^2=9 -1^2=1 0^2=0 -2^2=4 Step 4: Add the squared deviations. 9+1+0+4=14 Step 5: Divide the sum by the number of scores. 14/4=3.5 Step 6: Take the square root of the result from Step 5. square root of 3.5 ≈ 1.87 The standard deviation is approximately 1.87

Determine if research is correlational, experimental, or non-experimental research. What are the characteristics for each?

Correlational- measures a relationship between two variables without the researcher controlling either of them. It aims to find out whether there is either: Positive correlation or negative correlation. Experimental-type of research that uses a scientific approach towards manipulating one or more control variables and measuring their defect on the dependent variables Non-Experimental- he type of research that does not involve the manipulation of control or independent variable. In non-experimental research, researchers measure variables as they naturally occur without any further manipulation.

Know the definition of independent, dependent, and quasi-independent variables and recognize examples of each.

Dependent- the variable that is being measured or tested in an experiment. effect. Ex.in a study looking at how tutoring impacts test scores, the dependent variable would be the participants' test scores, since that is what is being measured. Independent- a variable that stands alone and isn't changed by the other variables you are trying to measure. The one that the researchers manipulate. Cause. Ex. amount of studying would be the independent variable Quasi-independent- any of the personal attributes, traits, or behaviors that are inseparable from an individual and cannot reasonably be manipulated. Ex. gender, age, and ethnicity.

Describe the effects of standardizing a distribution by transforming the entire set scores into z-scores,and explain advantages of this transformation.

Every score stays in the exact same position relative to every other score in the distribution. Mean - when raw scores are transformed into z-scores, the mean will always = 0.

Know the advantage of standardizing distributions is that two (or more) different distributions can be made the same.

For example, one distribution has μ=100 and σ=10, and another distribution has μ=40 and σ=6. When these distributions are transformed to z-scores, both will have μ=0 and σ=1.)

Know how the three types of frequency distribution graphs-histograms, polygons, and bar graphs are constructed and identify when each is used (for example, scale of measurement)

Histograms- are used to show distributions of variables, plot quantitative data with ranges of the data grouped into bins or intervals. no spaces between bars. Bar Graphs- used to compare things between different groups or to track changes over time, plot categorical data Polygon- constructed by using lines to join the midpoints of each interval, heights of the points represent the frequencies

Calculate variance and standard deviation for a simple set of scores.

How to find Variance- 1.subtract the mean from each score. 2.square each result. 3.sum all the square. 4.divide the sum of square by N. Standard Deviation- 1.solve for the mean (average) of the five test scores. 2.Subtract that mean from each of the five original test scores. 3.Square each of the differences. 4.Find the mean (average) of each of these differences you found in Step 2. 5.Take the square root of this final mean from #3. This is the standard deviation.

Know the definition of mean, median and mode and when each should be used (what are the advantages and disadvantages of each?)

Mean- (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set advantage- takes account of all values to calculate the average disadvantage- Very small or very large values can affect the mean. Median- the middle value when a data set is ordered from least to greatest advantage- not affected by very large or very small values disadvantage- , the item values have to be arranged, it is a less representative average because it does not depend on all the items in the series. Mode- the number that occurs most often in a data set. advantage- not affected by extremely large or small values, and can be located just by inspection in ungrouped data and discrete frequency distribution disadvantage- not well defined, and not based on all the values.

Define the median, identify the median for discrete scores, and calculate the precise median for continuous variables.

Median Discrete scores: the middle value when a data set is ordered from least to greatest Continuous scores: Median=L+(n2 − c.f.)f×i L= Lower limit of median class, median class is that class where nth/2 item is lying. c.f. = Cumulative frequency of the class preceding the median class. f = Frequency of median class. i = Class interval of median class.

Know the four scales of measurement (nominal, ordinal, interval, and ratio) and identify examples of each.

Nominal- used for naming or labelling variables, without any quantitative value ex. country, gender, race, hair color Ordinal- which the values follow a natural order, unlike interval or ratio data, ordinal data cannot be manipulated using mathematical operators. ex. socio economic status ("low income","middle income","high income"), education level ("high school","BS","MS","PhD") Interval- measured along a scale, in which each point is placed at an equal distance (interval) from one another. ex. the data collected on a thermometer—its gradation or markings are equidistant. Ratio- is defined as quantitative data, having the same properties as interval data, with an equal and definitive ratio between each data and absolute "zero" being treated as a point of origin, there can be no negative ex. time is ratio since 0 time is meaningful.

Understand percentiles and percentile ranks.

Percentiles- (or a centile) is a score below which a given percentage of scores in its frequency distribution falls. Percentile Ranks- describes the percentage of people in the comparison group who scored below a particular score. Ex- if a student's raw score of 62 corresponds to a percentile rank of 98, that student performed better than 98% of the other test-takers.

Using either the z-score definition or the z-score formula ,transform X values into z-scores and transform z-scores into X values for both populations and samples.

Population- Find Z score z= x-u/o The numerator measures the distance in points between X and μ and indicates whether X is located above or below the mean. The deviation score is then divided by σ because we want the z-score to measure distance in terms of standard deviation units. Sample- z= x-m/s Find X score Population- X= u+zo Sample- X=m+zs

Identify the shape of a distribution-symmetrical, positively or negatively skewed-by looking at a frequency distribution table or graph.

Positively Skewed- right side (or "tail") is longer than its left side Negatively Skewed- left side (or "tail") is longer than its right side Symmetrical- if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other

Be able to calculate the range as a simple measure of variability and interquartile range.

Simple Measure- Greatest number - lowest number Interquartile Range- Order the data from least to greatest. Find the median. Calculate the median of both the lower and upper half of the data. The IQR is the difference between the upper and lower medians.

.Calculate SS, the sum of the squared deviations, for population.

Step 1: Calculate the Sample Mean Step 2:Subtract the Mean From the Individual Values Step 3: Square the Individual Variations Step 4: Add the the Squares of the Deviations

Know how to find a 5 number summary boxplot.

Step 1: Put your numbers in ascending order (from smallest to largest). For this particular data set, the order is:Example: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27. Step 2: Find the minimum and maximum for your data set. Now that your numbers are in order, this should be easy to spot.In the example in step 1, the minimum (the smallest number) is 1 and the maximum (the largest number) is 27. Step 3: Find the median. The median is the middle number. If you aren't sure how to find the median, see: How to find the mean mode and median. Step 4: Place parentheses around the numbers above and below the median. (This is not technically necessary, but it makes Q1 and Q3 easier to find).(1, 2, 5, 6, 7), 9, (12, 15, 18, 19, 27). Step 5: Find Q1 and Q3. Q1 can be thought of as a median in the lower half of the data, and Q3 can be thought of as a median for the upper half of data.(1, 2, 5, 6, 7), 9, ( 12, 15,18,19,27). Step 6: Write down your summary found in the above steps.minimum = 1, Q1 = 5, median = 9, Q3 = 18, and maximum = 27.

Explain how the three measures of central tendency-mean, median, and mode are related to each other for symmetrical and skewed distributions, and predict their relative value based on the shape of the distribution

Symmetrical frequency curve-then mean, median and mode will be equal Positively skewed frequency distribution- the mean is always greater than median and the median is always greater than the mode. Negatively skewed- If the mean is less than the mode, the distribution is negatively skewed.

Calculate the probability of a specific outcome as a proportion,decimal,and percentage.

There is a 2/10, .2, 20% chance I will grab a red marble.

Know that an event is considered to be unusual if the probability of occurring is less than or equal to 0.05(or 5%).

Unusual Event has less that a 5% (.05) likelihood of happening

Define variance and standard deviation and describe what is measured by each.

Variance- is the average squared deviations from the mean. A measure of how spread out a data set is. It is calculated as the average squared deviation of each number from the mean of a data set. For example, for the numbers 1, 2, and 3 the mean is 2 and the variance is 0.667 Standard Deviation- is the square root of the variance. Standard deviation is expressed in the same units as the original values (e.g., minutes or meters).

Know z=0 is in the center (at the mean), and the extreme tails correspond to z-scores of approximately-2.00 on the left and +2.00 on the right.

Z=0 is Mean and Center EXTREME TAILS -2.00 on the left +2.00 on the right

Explain how z-scores establish a relationship among X, the mean, the standard deviation, and the value of z, and use that relationship to find an unknown mean when given z-score, and standard deviation; or find an unknown standard deviation when given a z-score, score, and the mean.

calculating a z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

Know the definition of descriptive and inferential statistics and how they're used, summarize and make decisions about data.

descriptive statistics focus on describing the visible characteristics of a dataset inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

Know the definition of discrete and continuous variables and identify examples for each.

discrete variables are countable in a finite amount of time. ex. home runs hit in a game A continuous variable is a variable whose value is obtained by measuring ex. height, weight, temperature, and time

Explain how a z-score identifies a precise location in a distribution for either a population or a sample of scores.

exact location in a distribution with a single number is the sign of z-score describes the location is above the mean "plus sign" and location below the mean "minus sign" and the numerical value of z-score specifies the distance from the mean in terms of number of standard deviations. Each z-score tells the exact location of the original X value within the distribution.