test #2 - stats and methods - ch. 3 & 4
symmetry
characteristic of having the same shape on both sides of the center
An advantage of the mean is
It can be manipulated algebraically
stem
vertical axis of display containing the leading digits
Scales of measurement
Nominal; Ordinal; Interval; Ratio
Median location
The location of the median in an ordered series
Sample form
divided by (n-1)
exploratory data analysis (EDA)
set of techniques developed by Tukey for presenting data in visually meaningful ways
Real limits
Denoted as "60-64" actually has "real limits" of 59.5-64.5.
Three important things about stem and leaf displays:
*They can be used to present both the shape of a distribution and the actual values of the scores. *they can be used back to back to compare two related distributions * They can be adjusted to handle different sized values for the dependent variables.
Positively skewed
- A distribution is positively skewed when is has a tail extending out to the right (larger numbers) When a distribution is positively skewed, the mean is greater than the median reflecting the fact that the mean is sensitive to each score in the distribution and is subject to large shifts when the sample is small and contains extreme scores.
Negatively skewed
- A negatively skewed distribution has an extended tail pointing to the left (smaller numbers) and reflects bunching of numbers in the upper part of the distribution with fewer scores at the lower end of the measurement scale.
Mesokurtic
- A normal distribution is called mesokurtic. The tails of a mesokurtic distribution are neither too thin or too thick, and there are neither too many or too few scores in the center of the distribution.
Leptokurtic
- If you move scores from shoulders of a mesokurtic distribution into the center and tails of a distribution, the result is a peaked distribution with thick tails. This shape is referred to as leptokurtic.
Platykurtic
- Starting with a mesokurtic distribution and moving scores from both the center and tails into the shoulders, the distribution flattens out and is referred to as platykurtic.
Outliers
-An extreme score that is not typical of the rest of the distribution -It may be larger than the other numbers or smaller than the other numbers. -Distorts the mean To find an outlier -Organize your data -Look for extreme scores -If the mean and median differ by a large amount, you have an outlier
Assume that you have a set of data with 70 values spread fairly evenly between 0 and 100. The optimal number of categories for a histogram of these data would be approximately:
10
"5s" represents what numbers on a stem-and-leaf display according to Tukey?
56-57
Mean, a visual representation
A visual point that perfectly balances two sides of a distribution.
Nominal
Assign numbers to objects where different numbers indicate different objects [1=male; 2=female]
Normal Distribution
Can have different means or standard deviations All normal distributions the same shape This means that a z-score will fall in the same relative location for different distributions STANDARD normal distribution always has mean 0, SD 1
Calculate the Mean
Forumula M= ΣX/N Step 1. Add up all the scores in a sample. Step 2. Divide the total of all the scores by the total number of scores.
Given the following data, what is the end of the whisker on the left? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
From the 1st quartile, go .90 to the left, which is 1.13 - .90 = 0.23 There is not a 0.23 in the data so move to the right until you reach 0.95
Bar Graph
Height of bar indicates frequency of occurrence; bars don't touch
Given the following data, what is the interquartile range? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
IQR = 3rd quartile minus 1st quartile = 1.73 - 1.13 = .60
Symmetrical unimodal
In a perfectly symmetrical unimodal distribution, mean, median, and mode are identical.
Given the following data, what is the median location? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
ML = (n+1) / 2 = (15 + 1) / 2 = 16 / 2 = 8
Given the following data, what is the maximum length of the whisker? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
Max length of whisker = 1.5 + IQR = 1.5 * .60 = .90
Given the following data, what is the end of the maximum whisker length? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
Maximum whisker length is 1.5 * IQR = 1.5 * 8 = 12
Central Tendency of a skewed distribution
Mean Median Mode SIQR; find values most typical of a group
Ratio
Order matters, intervals are equal and true zero point (absence of the property)
Descriptive Statistics
Statistical procedures used to summarize and describe the data from a sample. -Describe raw data with a single number -Way of capturing trends in data -Two Types of Descriptive Statistics
Mean
Sum of the observations divided by the number of observations = average. The average result of a test, survey, or experiment.
Three different terms for describing the shape of a distribution:
Symmetry, modality and skewness
Dependent
The change that happens because of the independent variable
Which measure of Central tendency is best?
The choice is usually between the mean and the median. The mean usually wins, but when distributions are skewed by outliers, the median may provide a better sense of a distributions central tendency.
Range
The difference between the largest and smallest data value in a data set
Deviation
The difference of a score in a set of scores from the mean of that set of scores.
Given the following data, list the outliers. .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
The ends of the whiskers are the highest and lowest numbers in the data. That means there are no outliers.
Suppose you wanted to construct a box plot of the following data. What would be the median? We will use this same data for the rest of the questions in this practice test. 50, 60, 75, 79, 80, 81, 82, 83, 84, 85, 87, 88, 92, 95, 100, 106, 120
The median location is 17 +1 divided by 2 equals 9 and the ninth number is 84
Median
The middle number or center value of a set of data in which all the data are arranged in sequence
MAD
The mean absolute deviation is the mean (average) of the absolute value of the difference between the individual values in the dataset and the mean. The method tries to measure the average distances between the values in the dataset and the mean.
Mean, in plain arithmetic
The mean is calculated by summing all the scores in a data set, then dividing the sum by the total number of scores.
Skewed Distribution
The mode is at the peak of the curve, mean is closest to the tail and median is positioned between the mode and the mean. The median is the best measure of central tendency for skewed distributions.
Mode
The mode is used in three situations: 1. When one particular score dominates a distribution. 2. When the distribution is bimodal or multimodal. 3. When data is nominal. When not sure which is best, report all three.
Mode, most common score
The most common scores of a sample from a frequency table, a histogram, or a frequency polygon. When there is more then one common score, and when scores have several decimal places it may be reported as a common interval i.e. 60-70.
Mode (Mo)
The most commonly occurring score
Median (Mdn)
The score corresponding to the point having 50% of the observations below it when the observations are arranged in numerical order
Mean (X-bar)
The sum of the scores divided by the number of scores
why do we use trimmed samples?
To eliminate the influence of extreme scores.
Shape of Distribution of Scores
Unimodal and perfectly symmetrical distribution Mean=Medain=Mode Skewed Distribution Mode> Median > Median negatively skewed Mode< Median< Mean positively skewed
Which of the following distributions can be symmetric?
Unimodal, normal, bimodal-all of these choices
Bimodal
When a distribution has two modes.
Unimodal
When a distribution of scores has more than one mode.
Multimodal
When a distribution of scores has more than two modes.
when is the median most useful?
When we don't want extreme scores to influence the result.
o Qualitative Variable
a variable based on categorical data.
Discrete Variable
a variable with a limited number of values (e.g., gender (male/female), college class (freshman/sophomore/junior/senior).
Correlational research
groups selected by the researcher; be cautious about making causal conclusions ○ The difference between the two - in correlational research, you cannot talk about causality
Mean, most common measure of Central Tendency
the arithmetic average of a group of scores. Often called the average and used to represent the typical score in a distribution as a precise calculation.
Population
the entire group that you are interested in; usually large
The endpoints of an interval are called ___
the real upper (and lower) limits
σ2 (population parameter)
variance - variance measures how far a set of numbers are spread out. A variance of zero indicates that all the values are identical.
When we make implicit assumptions about a scale having interval properties,
we are assuming the distance between 4 and 6 is the same as the distance between 6 and 8.
In deciding on the number of stems to use in a stem and leaf display,
you should normally make all of the stems the same width.
Variance
• subtract the mean from each of the values in the data set • square the result • add all of these squares • and divide by the number of values in the data set.
What is the real upper limit of the interval .80 - .89 in the table below? Please use 3 decimal places in your answer. .50 - .59 .60 - .69 .70 - .79 .80 - .89
0.895
What is the mode of the following distribution? Round your answer to the nearest 2 decimal places. 2 2 2 2 3 5 7 9 10 10 10
2
Calculate the 20% winsorized trimmed mean of the following distribution. Round your answer to the nearest 2 decimal places. x 10 13 14 17 19 22 24 24 26 27 27 27 28 29 30 37 47 77 88 100
27.2 replace these 4 and those 4 with the adjacent value
Given the numbers 6,7,9,11,15,71,86, how many numbers fall below the median?
3
correlation
A measure of the extent to which two factors vary together, and thus of how well either factor predicts the other. a reciprocal connection between two or more things
Normal distribution
A normal distribution of data means that most of the examples in a set of data are close to the "average," while relatively few examples tend to one extreme or the other
What is the 20% trimmed mean of the following distribution? Round your answer to the nearest 2 decimal places. X 10 13 14 17 19 22 24 24 26 27 27 27 28 29 30 37 47 77 88 100
The mean of the 20 numbers is 34.3 and the 20% trimmed mean is 26.67.
Sampling Distribution of the Mean
The mean of the sampling distribution of the mean is the same as the population mean The standard deviation of the sampling distribution of the mean: standard error of the mean < the standard deviation of the population distribution Extreme scores are more likely than extreme means, so the distribution of means will be less variable than the population As N increases, sample means are clustered more closely and the standard error gets smaller
Trimmed mean
The mean that results from trimming away (or discarding) a fixed percentage of the extreme observations
Degrees of Freedom (df)
The number of scores that are free to vary (N-1) Example: Out of the five scores, the first 4 scores can be anything. The 5th one's determined.
Independent
The one thing you change in the experiment; the variable you are observing and manipulating
Measurement data
sometimes called quantitative data -- the result of using some instrument to measure something (e.g., test score, weight);
s (sample statistics)
standard deviation
σ (population parameter)
standard deviation
Center of distribution
(1) Mode (2) Medium (3) Mean
Measure of dispersion
(1) Range (2) Mean Absolute Deviation (MAD) (3) Variance (4) Standard Deviation (5) SIQR - Semi-interquartile range
Shape of distribution
(1) Symmetric [one peak - unimodal] (2) Bimodal [two peaks] (3) Skew [skew left - more data to the right - left tailed] (4) Uniform
Dealing with Outliers
(1) Trimming - Winsorizing Trimmed mean - robust estimator of location (2) Data transformation - reduce the impact of outliers and make distribution more symmetric
Statistic
(Italic) M is a statistic, where numbers are based on a sample taken from a populations
Sum of Squares
(SS) A numerical value obtained by subtracting the mean of a distribution from each score in the distribution, squaring each difference, and then summing the differences.
Mean
(X) The sum of set of scores divided by the number of scores summed. - Arithmetic average -Most common measurement of central tendency -Influenced by extreme scores -Data should have interval properties; can not be used with nominal or ordinal data -Sample mean is the best estimator of population mean. -Can be manipulated algebraically. -Any change of a score in the distribution affects the mean
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the specificity of the measurement tool?
0.69
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the sensitivity of the measurement tool?
0.78
What is the real lower limit of the interval .60 - .69 in the table below? Please use 3 decimal places in your answer. .50 - .59 .60 - .69 .70 - .79 .80 - .89
0.595
Parameter
(mew μ) A number based on the whole population; parameters are usually symbolized by Greek letters.
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the Prevalence of disease in the sample?
.26
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the positive predictive value of the measurement tool?
.47
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the efficiency of the screening instrument?
.71 -- Efficiency is all true tests divided by total sample. That is (125/175) x 100 = .71428 x 100 = 71.43
A measurement instrument was used at Mercy Hospital in a sample of 175 patients. There were 35 true positives, 40 false positives, 10 false negatives and 90 true negatives. What is the negative predictive value of the meaurement tool?
.9
The real lower limit and the real upper limit of the interval 40-49 are:
39.5 and 49.5
For the following data set [1, 7, 9, 15, 33, 76, 103, 118], what is the median location?
4.5
For the following set of data , the mean is:
5
What is the mean of the following distribution? Round your answer to the nearest 2 decimal places. 2 4 6 7 8
5.4
Given the following data, list the outliers. 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
50, 60, 100. These are the values that go beyond the ends of the whiskers.
What is the real lower limit of the interval 60 - 69 in the table below? Please use 2 decimal places in your answer. 50 - 59 60 - 69 70 - 79 80 - 89
59.5
What is the median of the following distribution? Round your answer to the nearest 2 decimal places. 2 4 6 7 8
6
What is the median of the following distribution? What is the median of the following distribution? Round your answer to the nearest 2 decimal places. 2 4 6 6 6 10 20 25
6.17 -- RLL of 6 plus 2/3 5.5 + .67 = 6.17
What is the median of the following distribution? Round your answer to the nearest 2 decimal places. 2 4 6 7 8 10
6.5
Upper real limits
60 + 0.5 = 60.5/ 64 + 0.5 = 64.5
Lower real limits
60 - 0.5 = 59.5/ 64 - 0.5 = 63.5
Suppose you wanted to construct a box plot of the following data. What is the end of the lower whisker? This is the end of the lower whisker, not the maximum lower whisker. We will use this same data for the rest of the questions in this practice test. 50, 60, 75, 79, 80, 81, 82, 83, 84, 85, 87, 88, 92, 95, 100, 106, 120
75
Suppose you wanted to construct a box plot of the following data. What is the first quartile? We will use this same data for the rest of the questions in this practice test. 50, 60, 75, 79, 80, 81, 82, 83, 84, 85, 87, 88, 92, 95, 100, 106, 120
80
The mode of the numbers 1 3 4 5 6 6 7 8 9 9 9 is
9
The mode of the numbers 1,3,4,5,6,6,7,8,9,9,9 is
9
The standard normal distribution (z-score distribution)
A normal distribution with μ = 0 and σ = 1 The distribution you get when you transform all the scores from any normal distribution into z-scores Area under the curve equals 1 Area under the curve is the probability of an event
Sample
A portion of the population selected for a study
Random sample
A sample drawn in such a way that each element of the population has the same chance of being included in the sample
Ordinal
Assign numbers to objects, but here the numbers also have meaningful order. But spaces between an adjacent value are not necessarily equal [class ranking, small, medium, large, IQ]
Given the following data, what is the median location? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
Ml = (n+1) / 2, which equals (10 + 1) / 2 = 5.5
Histogram
Bar graph; bars do touch [real and apparent limits]
Apparent limits
Class intervals; the values denoting the interval as 60-64
Converting a raw score into a z-score (Linear Transformation)
Converting a set of raw scores into z scores will NOT change the shape of the distribution Ex: If I convert all the scores in a positively skewed distribution to z-scores, my resulting distribution will have a mean of 0, a standard deviation of 1, and it will be POSITIVELY SKEWED (the shape will not change)
Given the following data, what is the first quartile? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
Count over 3 from the left (3 is the QL) and you get 1.13
Parametric
statistical techniques that deal with looking at population parameters and their distributions of values
Given the following data, what is the whisker end on the right? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
Count over to the right, end of whisker plus max whisker length = 1073 + .90 = 2.63 There is not a 2.63 in the data so move to the left until you reach 2.03
Outliers
Data skewed by one or a few outliers that are extreme scores either very high or very low in comparison to other scores. When the outlier is omitted from the density data, the means becomes more representative of the actual scores in the sample.
Median, the middle score
Median is the middle score of all the scores in a sample when the scores are arranged in ascending order. Step 1. Line up all the scores in ascending order. Step 2. Find the middle score. Calculate the mean of the two middle scores if even numbers.
Median
Middle score -The score that has an equal number of scores above and below it (the 50th percentile). -It cuts the distribution into two equal parts. 50% split of data. -Not affected by extreme scores (desirable for skewed distributions). -Can be used with ordinal and interval data, but not with nominal data. -Does not take into account all scores. -Not a stable measure of central tendency.
Mode
Most frequent score Finding the Mode -Put the data in order -Choose the most frequent occurring score in the data set UNIMODAL: distribution has only one mode. BIMODAL: distribution has two modes MULTIMODAL: distribution has more than 2 modes. -Mode may not appear in all data sets. -Data set may contain multiple modes. -Not a stable measure of central tendency. -Not affected by extreme scores. -Can be used with nominal, ordinal interval, or ratio data.
Stem-and-Leaf
Most information about individual scores
Which measurement of Central Tendency to Use
Nominal Data -Mode Ordinal Data -Median response Interval or Ratio Data -Symmetrical Distribution (No outliers) -Mean Skewed Distribution (Outliers) -Median
Scale of Measurement of Scores
Nominal: Mode Ordinal: Mode Median Interval: Mode Median Mean Ratio: Mode Median Mean
SIQR
Not affected by outliers § Interquartile range - Q3-Q1 □ The medium of the first half of the distribution and the medium of the third half of that § Semi-interquartile range - interquartile range divided by 2 □ (Q3 - Q1)/2 Q1 25% upper real limit Q2 Medium Q3 75% upper real limit
Sample size
Number of units in a sample
Interval
Numbers have orders, but there are also equal intervals between adjacent values [temperature]
Measures of Variability
Numbers that indicate how much scores differ from each other and the measure of central tendency in a set of scores. -Range, Variance, Standard Deviation
Measures of Central Tendency
Numbers that represent the average or typical score obtained from measurements of a sample. -Indicate typical score obtained -Mean, Median, Mode
Measures of central tendency
Numerical values that refer to the center of the distribution
Given the following data, what is the quartile location? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
QL = (ML + 1) / 2 = (8+1) / 2 = 9/2 = 4.5 but you drop the fraction so it is 4
Given the following data, what is the quartile location? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
QL = (ML + 1) / 2, which equals (5 + 1) / 2, which equal 6/2 = 3 Note, we dropped the fraction in the ML, from 5.5 to 5
Given the following data, what is the interquartile range (IQR)? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
Quartile on the right is 4 digits over = 85 Quartile on the left is 4 digits over = 77 88 - 77 = 8
Relative Frequency and Cumulative Relative Frequency Distributions
RF (frequency divided by N) Can determine the fraction of scores at or below your score; CRF (cumulative frequency divided by N)
Central Tendency
Refers to the descriptive statistic that best represents the center of a data set, the particular value that all the other data seem to be gathering around. p. 79 The central tendency is usually at the highest point in the histogram.
Mean, symbolic notation
Several symbols represent the mean: M or x-bar For samples from a population M is a statistic. For a population the Greek letter μ (mew) Sigma or Σ is the summation symbol. N is the total number of scores in a data set n is the number of sample scores
Standard deviation
Standard deviation is the square root of the variance ○ If a constant is added to (or subtracted from) every score in a distribution, the standard deviation will not be affected ○ If every score is multiplied (or divided) by a constant, the standard deviation will be multiplied (or divided) by that constant ○ The standard deviation from the mean will be smaller than the standard deviation from any other § Measure of dispersion/ measure of spread
Given the following data, what is end of the whisker on the left side of the distribution? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
Start from the 1st quartile, which is 77 Subtract the maximum whisker length = 77 - 12 = 65. This is the maximum possible location of the whisker. There is not a 65 in the data so move to the right until you find the data that represents the end of the whisker, which is 73.
Given the following data, what is end of the whisker on the right side of the distribution? 50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
Start from the 3rd quartile, which is 85 Add the maximum whisker length = 85 + 12 = 97. This is the maximum possible location of the whisker. There is not a 97 in the data so move to the left until you find the data that represents the end of the whisker, which is 95.
Positively skewed distribution has a tail stretching out to the right
TRUE
On a recent fundraising drive, most of the 30 volunteers raised between $10 and $50 each. However, Brian and Karen each raised over $100. Which of the following is true
The amounts of money raised by Brian and Karen are outliers.
Properties of z-scores
The mean of a complete set of z-scores is 0 The standard deviation is 1 Converting a set of raw scores into z-scores will not change the shape of the distribution. You can use z-scores to find the location of one group with respect to all other groups of the same size
Mean, the arithmetic average
The mean is simple to calculate and a gateway to understanding statistical formulas.. It is an important concept in statistics with four ways to think about it: verbally, arithmetically, visually and symbolically (using statistical notation).
A figure that plots various values of the dependent variable on the X axis and the frequencies on the Y axis is called____
a histogram or frequency distribution to some
Sampling Distribution vs. Population Distributions
The sampling distribution is normal if the population distribution is normal The sampling distribution will approach normal even if the population distribution is not normal (if N is large enough - Central Limit Theorem) Mean will be the same as population distribution, but the variability is less (smaller standard error) Must use the sampling distribution of the mean for groups
Mode
The value or values that occur most frequently in a data set
How data is measured in Central Tendency
The way data clusters around central tendency is measured in three different ways: mean, median and mode
If the distribution of the ages of people were positively skewed, which of the following is most likely correct?
There are more young people than old people.
Discrete
a finite number of values and there are gaps between adjacent values [# of students] Ordinal + Nominal
Statistics
a set of concepts, rules, and procedures that help us to: o organize numerical information in the form of tables, graphs, and charts; o understand statistical techniques underlying decisions that affect our lives and well-being; and o make informed decisions.
Standard deviation
a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data. When the examples are pretty tightly bunched together and the bell-shaped curve is steep, the standard deviation is small. When the examples are spread apart and the bell curve is relatively flat, that tells you that you have a relatively large standard deviation. About 68% of the data will fall within one standard deviation of the mean, 95% of the data will fall within two standard deviations of the mean and 99.7% of the data will fall within three standard deviations of the mean.
Sample
a subset of a population which is too large to measure; the representative of the population
Continuous Variable
a variable that can take on many different values, in theory, any value between the lowest and highest points on the measurement scale.
Independent Variable
a variable that is manipulated, measured, or selected by the researcher as an antecedent condition to an observed behavior. In a hypothesized cause-and-effect relationship, the independent variable is the cause and the dependent variable is the outcome or effect.
Dependent Variable
a variable that is not under the experimenter's control -- the data. It is the variable that is observed and measured in response to the independent variable.
Mean
affected by outliers ○ If a constant is added to (or subtracted) from every score in a distribution, the mean is increased (or decreased) by that constant ○ If every score is multiplied (or divided) by a constant, the mean will be multiplied (or divided) by that constant ○ The sum of the deviations from the mean will always equal zero ○ 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 Symbolized in population by μ ("mu") Sample: M or "X bar"
For the data set [1, 3, 3, 5, 5, 5, 7, 7, 9], the value "5" is:
all of these choices:mode, median, mean
Categorical data
also referred to as frequency or qualitative data. Things are grouped according to some common property(ies) and the number of members of the group are recorded (e.g., males/females, vehicle type).
Continuous
an infinite number of values and there are no gaps between adjacent values [time] Ratio and Interval
diagram in which occurrence frequency of different values of X is represented by height
bar graph
Mode cannot
be calculated algebraically
a normal distribution must
be symmetric
A normal distribution must
be symmetric.
Which of the following is the least important characteristic of graphics?
beauty
The onset of eating disorders was shown to occur most often during puberty and during the late teen years in girls. A distribution of the frequencies of onset of eating disorders by age would most likely be:
bimodal.
Which of the following can be defined algebraically?
both mean and median location
real lower limit
boundary halfway between the bottom of one interval and the top of the next
real upper limit
boundary halfway between the top of one interval and the bottom of the next
The "real lower limit" of an interval in a histogram is
c. the lowest continuous value that would be rounded up into that interval.
unimodal
characteristic of distribution having one distinct peak
Frequency Polygon
connects the dots from the histogram
Given the following data, what is the median? .95, 1.06, 1.13, 1.40, 1.41, 1.56, 1.63, 1.73, 1.73, 2.03
count over to the rigth, 5.5 and the ML is between 1.41 and 1.46. The mean of these is 1.485
Descriptive Statistics
describe the data that you are looking at; for a sample and calculate inferential statistics to make inferences of the population
Histogram
diagram in which rectangles are used to represent recurrence of observations within each interval
line graph
diagram in which the Y values corresponding to different values of ? are connected
Mode
distinguish multimodal from unimodal distribution Unreliable, but the only measure of central tendency for nominal scales
Population form
divided by n
Outliers are
extreme or unusual values.
Data
facts, observations, and information that come from investigations.
stem-and-leaf display
graphic presenting original data arranged into a histogram
A negatively skewed distribution
has a tail pointing to the left
A negatively skewed distribution
has a tail pointing to the left.
leaf
horizontal axis of display containing the trailing digits
The primary purpose of plotting data is to make them___
interpretable
Which of the following is not an advantage of the median?
it can be manipulated algebraically
leading digit
leftmost numeral of a number
In using ordinal data, which measure of central tendency is probably least useful?
mean
When the distribution is symmetric, which of the following are always equal?
mean and median
When the distribution is symmetric and unimodal, which of the following are always equal?
mean, median, and mode
The measure of central tendency that is most useful in estimating population characteristics because it is less variable from sample to sample is the:
mean.
Skewness
measure of the degree to which a distribution is asymmetrical
Alison received a score of 480 on the verbal portion of her SAT. If she scored at the 50th percentile, her score represents the ________ of the distribution of all verbal SAT scores.
median
If a store manager wanted to stock the men's clothing department with shirts fitting the most men, which measure of central tendency of men's shirt sizes should be employed?
mode
On a histogram, which always refers to the highest point on the distribution?
mode
Which of the following is useful with data collected with nominal scales?
mode
Nonparametric
not looking at the distributions of those parameters
modality
number of major peaks in a distribution
less significant digit
numeral to the right of the leading digit
A major characteristic of a good graphic is___
simplicity
Standard deviation
o - (s or ) is defined as the positive square root of the variance. The variance is a measure in squared units and has little meaning with respect to the data. Thus, the standard deviation is a measure of variability expressed in the same units as the data. The standard deviation is very much like a mean or an "average" of these deviations. In a normal (symmetric and mound-shaped) distribution, about two-thirds of the scores fall between +1 and -1 standard deviations from the mean and the standard deviation is approximately 1/4 of the range in small samples (N < 30) and 1/5 to 1/6 of the range in large samples (N > 100).
Symmetric
o - Distributions that have the same shape on both sides of the center are called symmetric. A symmetric distribution with only one peak is referred to as a normal distribution.
Kurtosis
o - Like skewness, kurtosis has a specific mathematical definition, but generally it refers to how scores are concentrated in the center of the distribution, the upper and lower tails (ends), and the shoulders (between the center and tails) of a distribution.
Interquartile Range (IQR)
o - Provides a measure of the spread of the middle 50% of the scores. The IQR is defined as the 75th percentile - the 25th percentile. The interquartile range plays an important role in the graphical method known as the boxplot. The advantage of using the IQR is that it is easy to compute and extreme scores in the distribution have much less impact but its strength is also a weakness in that it suffers as a measure of variability because it discards too much data. Researchers want to study variability while eliminating scores that are likely to be accidents. The boxplot allows for this for this distinction and is an important tool for exploring data.
Skewness
o - Refers to the degree of asymmetry in a distribution. Asymmetry often reflects extreme scores in a distribution.
Variance
o - The variance is a measure based on the deviations of individual scores from the mean. As noted in the definition of the mean, however, simply summing the deviations will result in a value of 0. To get around this problem the variance is based on squared deviations of scores about the mean. When the deviations are squared, the rank order and relative distance of scores in the distribution is preserved while negative values are eliminated. Then to control for the number of subjects in the distribution, the sum of the squared deviations, (X - X), is divided by N (population) or by N - 1 (sample). The result is the average of the sum of the squared deviations and it is called the variance.
Histogram
o - a form of a bar graph used with interval or ratio-scaled data. Unlike the bar graph, bars in a histogram touch with the width of the bars defined by the upper and lower limits of the interval. The measurement scale is continuous, so the lower limit of any one interval is also the upper limit of the previous interval.
Scatterplot
o - a form of graph that presents information from a bivariate distribution. In a scatterplot, each subject in an experimental study is represented by a single point in two-dimensional space. The underlying scale of measurement for both variables is continuous (measurement data). This is one of the most useful techniques for gaining insight into the relationship between tw variables.
Bar graph
o - a form of graph that uses bars separated by an arbitrary amount of space to represent how often elements within a category occur. The higher the bar, the higher the frequency of occurrence. The underlying measurement scale is discrete (nominal or ordinal-scale data), not continuous.
Boxplot
o - a graphical representation of dispersions and extreme scores. Represented in this graphic are minimum, maximum, and quartile scores in the form of a box with "whiskers." The box includes the range of scores falling into the middle 50% of the distribution (Inter Quartile Range = 75th percentile - 25th percentile)and the whiskers are lines extended to the minimum and maximum scores in the distribution or to mathematically defined (+/-1.5*IQR) upper and lower fences.
Quantitative Variable
o - a variable based on quantitative data.
frequency distribution
occurrence in which dependent variable values are tables or plotted against their recurrence
Cumulative Percentage polygon
one can estimate the percentile rank by simply looking at the graph
Percentile
percentiles are upper real limits of a category. Thus, the 95th percentile is X=4.5, the upper real limit of the X=4 category.
"u" is the
population mean
μ (population parameter)
population mean/ mean of population values
Variable
property of an object or event that can take on different values. For example, college major is a variable that takes on values like mathematics, computer science, English, psychology, etc.
To get an accurate idea about the shape of a distribution:
relatively large samples of data are needed.
Which of the following is an advantage of the median?
relatively unaffected by extreme scores it does not depend on the assumption of interval or ratio level data
trailing digit
rightmost numeral of a number
X(with a '-' on the top of it) sample statistics
sample mean
Xbar is the
sample mean
Someone asks you if you have seen the movie Titanic. Before you answer, you look back into your memory for all of the movies you have ever seen and review the titles one at a time. This is an example of
sequential processing
If the mean score of test #1 was 80.00 in section 01 with 20 students, 70.00 in section 02 with 15 students and 50.00 in section 03 with 40 students, what is the mean score of all students in all three sections? Round your answer to the nearest 2 decimal places.
sum of mean*n = 4650 sum of n = 75 4650 / 75 = 62.00 Below is how I answered the question with Excel. nmean n * mean Section 012080 1600 Section 021570 1050 Section 034050 2000 sums =75200 4650 62 Weighted mean = sum of (n * mean) divided by sum of n, which is 4650 / 75 in this example
What is the mean of the following frequency distribution? Round your answer to the nearest 2 decimal places. X f 2 5 3 6 4 4
sum of xf = 44 sum of f = 15 44 / 15 = 2.93
The Cumulative Frequency Distribution
sums the frequencies at and below a particular value; determine the number of scores at or below your score
Inferential Statistics
takes the numbers to make inferences to a big population
Range
the difference between the largest value and the smallest value in the data set; unreliable because it is only based on two scores in the dataset
The "real lower limit" of an interval in a histogram is
the lowest continuous value that would be rounded up into that interval.
which of the measures of central tendency are you most likely to see reported in the popular press?
the mean
give two advantages of the mean relative to the other measures
the mean gives a more stable estimate of the central tendency of a population over repeated sampling. the mean can be used algebraically.
Positive Skew
the mean is greater than the median Ceiling Effect - bounce the distribution at the top end (maximum score)
Negative Skew
the mean is less than the median Floor Effect - bounce the distribution at the lower end (minimum score)
If we were interested in studying salaries in the National Basketball Association, the least useful measure of the typical salary would be
the mean.
We are most likely to randomly pick which score from an actual data set?
the mode
An advantage of the mode is
the mode can be used with nominal data
If you created a stem-and-leaf display of the math SAT scores of all entering students in a large Midwestern state university, the stem would best be:
the numbers 2 through 8.
Parameters
the numbers that we calculate or measure from a population; noted as Greek letter; characteristic of population example: population mean is a type of parameter
Percentile rank
the percentile rank of X=3.5 is 70% (note that 3.5 is the upper real limit of the category where X=3)
Central linear theorm
the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough
Medium
the second quartile (divide the distribution into 4 parts); 50th percentile not affected by outliers because they do not rely on extreme scores
Cumulative Frequency Polygon
the shape of the graph is a common shape called ogire
The optimal number of intervals for a histogram (and for a stem and leaf display) is____
whatever makes the figure show the most useful description of the data without creating too many or too few intervals.
Skewed distribution
§ The means is pulled in the direction of the skewed, most misleading § Use the medium and SIQR as the central tendency
Measures of Shape
• - For distributions summarizing data from continuous measurement scales, statistics can be used to describe how the distribution rises and drops.
Measures of Center
• - Plotting data in a frequency distribution shows the general shape of the distribution and gives a general sense of how the numbers are bunched. Several statistics can be used to represent the "center" of the distribution. These statistics are commonly referred to as measures of central tendency.
Graphs
• - visual display of data used to present frequency distributions so that the shape of the distribution can easily be seen.
