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Terms in this set (70)
numbers computed from data that describe the data's central tendency, variability, and other characteristics
most frequently used measure of central tendency. Subscripts can be used to represent means of various groups, mean of population is represented by u
mean is calculated by
Adding all values of the variable x and divided by the number of those values
Will the means computed from ungrouped frequency distributions and raw scores be identical?
Can means be computed from grouped frequency distributions?
Yes but they are only an estimate, the wider the interval the more variability we can expect
what is the grand mean?
The grand mean (weighted mean) is computed by combining the means of two or more groups to obtain an overall mean
What point is the mean?
The mean is the point in the distribution around which all the deviations sum to zero
What is unique about the sum of the squared deviations around the mean?
It is smaller than the sum of deviations around any other number, least squares criterian.
What is a downside of the mean?
It is strongly influenced by outliers and is pulled in their direction. Can also be misleading in a bimodal distribution because scores are either higher or lower than the mean.
used less frequently than mean, N+1/2 score, arrange scores in ascending or descending order, easily found in ungrouped frequency distributions
Is the median the score at the 50th percentile
No, that is false
Is the median the score at which 50% of scores fall above and below?
No, that is false
The median is the point at which...
Equal number of scores fall above and below
What is a positive about the median?
It is less affected by extreme scores, is preferable to the mean when simply measuring central tendency.
Negatives about the median
Is not resistant to the affects of bimodal data.
When do the mean and median occur at the same point?
In non-skewed distributions
What is the mode?
The score having the greatest frequency of occurance
How is the mode computed?
By counting, in a grouped freq distribution it is the midpoint of the interval with the greatest frequency.
What types of scales is the mean used for?
Interval and ratio scales, also dichotomously scored nominal variables
What types of scales is the median used for?
interval and ratio only
What is the mode used for?
interval, ratio and multicategorical nominal variables
what do descriptors of variablity do?
They make it possible to describe a characteristic of data with some degree of precision
Highest score minus the lowest, least stable variability measure, influenced by outliers
difference between scores at the 75th and 25th percentiles, not influenced by outliers
averaged squared deviation of scores around the mean Sigma(X-mean)2/N
How does variance change?
Variance changes as a function to the amount of variability in the data, when all scores are equal variance =0
used to estimate the population variance, N-2 as the denominator, larger sample sizes show smaller differences between variance and corrected variance
Sum of squares
sum of squared deviations around the mean sigma(X-mean)2 as the N increases so does the SS
the square root of the variance, makes the variance easier to understand. Equal to the average deviation of scores around the mean s= square root of s2
coefficient of variation
used to compare the variability of scores obtained by different measuring procedures. V = s/median
Can the variance be used for nominal scales
Sort of, it can be used for dichotomously scored nominal variables
sk= mean-median/median, neg skewed= neg value, pos skewed = pos value
sigma (x-mean)4/n/(s2)2, platykurtoic is less than 3, leptokurtic is greater than 3, mesokurtic =3
are called z-scores, transformation of raw scores that specify each score's amount and direction of deviation from the mean
z-score tells us what?
How many standard deviations fall between the raw score and the mean. zx = x -mean/s
z scores tell us 2 things
1) the sign (positive or negative) shows the direction of the score's deviation 2) absolute magnitue of scores tells us how many standard deviations fall between a score and the mean.
Any distribution of standard scores will show 3 characteristics
1) the mean of the distribution of z scores will always be zero 2) the variance and std dev are always equal to 1. 3) standardizing a distribution does not alter the shape of that distribution
What can standard scores help us locate?
They help us locate scores relative to the mean of the distribution. Standard scores maintain an interval level of measurement unlike percentiles.
What is the downside to z scores?
z scores are less easily interpreted.
What can we use to compare scores obtained using different measuring procedures?
Is bell shaped, 34.13% of cases fall between mean and 1 std deviation and 13.59% of cases fall between 1 and 2 std dev
What can we assume about distributions that look normal?
We can assume that they are close enough so that statistical conclusions based on the normal assumptions will be acceptable
standard normal distribution
Any normal distribution of standard scores 1) mean is always zero 2)bell shaped 34.13% between 0 and 1 and 13.59% between 1 and 2 and so on..
How do we apply standard normal distribution?
We can use the standard normal distribution to estimate percentiles, estimate raw scores corresponding to percentiles (which scores would fall into the 80th percentile?) and assigning cases to groups
Modified standard scores
Can be created to form a distribution with any desired mean and standard deviation mod z = s desired(z) + mean desired
normalized standard scores
normalizes a non-normal distribution of scores. First you look at cum percentages and corresponding z scores for each percentile
normalization is generally only recommended for..
In cases in which deviations from the normal distribution are due to a deficiency in the procedure used to measure the variable being studied.
assigned to normal distribution of scores- bottom 4% -1 next 7% 2 next 12% 3, etc. elminates the negative values, but there is a loss of precision
theoretical frequency distributions that depcit the frequency of occurrence of values of some statistic computed for all samples of size N
Facts about sampling distributions
They are imaginary entities, sampling distributions do not display frequency of occurance of scores on some variable
sampling distribution of the mean
formed by drawing all possible samples of size N from a population, computing the mean for each sample, and then plotting the frequency of occurance of each of the various values of the mean obtained from one sample to the next
how do sampling distributions differ from one to the next?
1) whether single samples or multi-sample sets are being drawn 2) what stat is being computed 3) sample size
the variability in sample means seen from one sample to the next
why is it called sampling error?
Because no sample is likely to match the population from which it was drawn in every way. The sample is said to be "in error"
How are sampling distributions formed?
By using sampling with replacement
What does the central limit theorem tell us?
1) sampling distribution of the mean will be approximately normal in shape and become increasingly normal as N increase 2) mean of distribution will equal the population mean 3) the standard error can be computed as omean= o/square root of N of the mean will always be less than the standard deviation of the population distribution
what's another name for the standard error of the mean
the standard deviation of the sample distribution of the mean
How does the size of the sample effect the standard error of the mean?
The larger the sample size the lower the standard error of the mean will be
Is the sampling distribution of the proportion different from the sampling distribution of the mean?
No it's basically the same thing and the same aspects of the central limit thereom can be applied.
When we are dealing with proportions what are we really dealing with?
We are really dealing with means of dichotomously scored variables
What are the two approaches for estimating population parameters from sample statistics?
1) point estimation 2) interval estimation
point estimation uses a sample-based statistic as the single value that best estimates the corresponding population parameter.
uses sample data to compute a range or "interval" of values which has a known probability of capturing the population parameter - called a confidence interval. Most commonly used to estimate population means and population proportions or percentages.
confidence interval for population mean
mean +/- 1.96 (standard error of mean) (same for population proportion confidence interval)
width of confidence intervals
all other things being held equal, narrow intervals are preferable to wide intervals
three things that determine interval width
1) level of confidence 2) amount of variability in data 3) sample size
How does the confidence interval effect the interval width
the higher the level of confidence the wider the interval
How does the variability effect the interval width
The more variable the data the wider the interval, only way this can be controlled is to use measuring procedures and instruments that are as reliable as possible
Sample size effects interval width by
The smaller the sample size the greater the standard error and the wider the confidence interval. A narrower interval may be purchased by gathering data from more cases.
how to choose practical sample size
1) desired level of confidence 2) acceptable width 3) approx variability
Recommended textbook explanations
Mathematical Statistics with Applications
Dennis Wackerly, Richard L. Scheaffer, William Mendenhall
A First Course in Probability
A Survey of Mathematics with Applications
Allen R. Angel, Christine D. Abbott, Dennis C. Runde
Statistical Techniques in Business and Economics
Douglas A. Lind, Samuel A. Wathen, William G. Marchal
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