Statistics Chapter 4
what does the standard deviation use, and what does it measure?
it uses the mean of the distribution as a reference point and it measures variability by considering the distance between each score and the mean
N=6 scores: 12, 0, 1, 7, 4, 6. locate the mean. calculate SS, variance and standard deviation.
mean = 5 SS= 96 Variance = 16 Standard Deviation = 4
what is MS?
mean square: often used to refer to variance, which is the mean squared deviation.
if distribution of scores are all the same, they are said to have _____. small differences between scores is ____. large differences between scores is ____
no variability. small variability. large variability.
the deviation scores are calculated for each individual in a population of N=4. the first 3 individuals have deviations of +2, +4 and -1. What is the deviation for the 4th individual?
-5
what is the equation for sample variance?
SS/ n-1
explain a biased VS unbiased sample statistic
a sample statistic is unbiased if the average value of the statistic is equal to the population parameter; the average value of the statistic is obtained from all the possible smiles for a specific sample size a sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter
find the sum of the squared deviations, SS, for each of the following populations: (note which formula works better for each population) a) 3, 1, 5, 1 b) 6, 4, 2, 0, 9, 3
a) computational formula works best SS= 11 b) definitional formula works best SS= 50
If μ= 50 and σ= 10. a) would a score of 58 be considered extreme? b) what if σ=3?
a) score of 58 is located in the central section of the distribution; within one standard deviation. b) with σ=3 a score of X=58 is an extreme value, located more then two standard deviations above the mean.
A population has a mean of μ=70 and SD of σ= 5. a) if 10 points were added to every score in the population, what would be the new values for the population mean and standard deviation? b) if every score in the population were multiplied by 2, what would be the new values for the population mean and standard deviation?
a) μ= 80. SD would remain the same; σ=5. b) μ= 140. SD σ= 10
you can check calculations when working with deviation scores by:
checking to see if the deviation scores add up to zero. A complete set of deviation will always add up to zero. The mean of the deviations is also zero.
what does the + or - mean when calculating deviation scores?
the direction from the mean: whether the score is located above (+) the mean or below the mean (-)
deviation is:
the distance from the mean
what is the deviation?
the distance from the mean for each score.
population variance = variance =
the mean squared deviation variance is the average squared distance from the mean.
what describes whether the scores are spread out or clustered together?
variability
variance equation: standard deviation equation:
variance = SS/N standard deviation = √SS/N
explain why the formula for sample variance divides SS by n-1 instead of dividing by n
without some correction, sample variability consistently underestimates the population variability. Dividing by a smaller number (n-1) increases the value of the sample variance and makes it an unbiased estimate of the population variance.
standard deviation =
√variance
what is the difference between population and sample variance?
population variance = SS/ N sample variance= SS/n-1
σ2 . what does this mean?
population variance. you would square root this to get the population standard deviation.
M= .... μ = .... N=..... n= .....
M= mean of sample μ = mean of population N=number of scores in population n= number of scores in sample
n=5 scores: 3, 1, 9, 4, 3 locate mean. calculate SS, variance and standard deviation for this sample.
M=4 SS=36 Variance = 9 Standard deviation = 3
μ =
population mean
what are the problems with using range as a measurement of variability?
1) it does not take into account scores in the distribution aside from the two extreme values. 2) because it does not consider all scores, it often doesn't give an accurate description of the variability for an entire distribution
how do you compute the standard deviation in a population?
1. Calculate the mean: ∑X/N 2. Once you have the mean, establish the deviation: (X - μ) 3. Make new column in table: (X-μ) . these are the deviation scores 4. Next, square each deviation 5. find the average of the squared deviations (called variance) 6. take the square root of the variance
when computing equations to find standard deviation, what are the equations in order from start to finish?
1. calculate mean (sum of X, divide by N) 2. use mean to compute SS (either definitional or computational formula) 3. use SS to compute variance (divide SS by N) 4. use variance to calculate standard deviation (square root of variance to find standard deviation)
what are the two different formulas to calculate sum of squares?
1. definitional formula: SS = ∑(x-μ)2 2. computational formula: SS = ∑X2 - (∑X)2/N
a good measure of variability will serve two purposes:
1. describes the distribution; if the scores are clustered together, or spread out over a large distance. 2. measures how well the individual scores (or group of scores) represent the entire distribution; variability provides information about how much error to expect if you are using a sample to represent a population
in frequency distribution graphs we identify the standard deviation by:
a line or arrow drawn from the mean outward for a distance equal to the standard deviation and labeled with an "σ" or "s"
Set of scores: 1, 5, 7, 3, 4 a) N=5 compute SS and variance b) n=5 compute SS and variance
a) SS = 20 variance = 4 b) SS= 20 variance = 5
a complete set of deviation always
adds up to zero
what happens to standard deviation if you add a constant to each score, or multiply each score by a constant?
all scores are increased/ all scores are multiplied; however the deviation score does not change.
what is the deviation score equation?
deviation score = X - μ (population mean)
for a sample of n scores, the degree of freedom (or df) for the sample variation are defined as
df= n-1
in frequency distribution graphs we identify the position of the mean by:
drawing a vertical line and labelling it with μ or M
what else is a sample standard deviation called?
estimated population standard deviation
what else is a sample variance called?
estimated population variance
where is the only computational difference when computing samples as opposed to populations?
in the part where we are calculating population/sample variance. sample: SS/n-1 population: SS/N
the mean, the standard deviation and the variance should be used only with ___ _____ from ____ ______ scales of measurement.
numerical scores from interval or ordinal scales of measurement
σ = ? 2 options
population standard deviation SS/N or √σ2
what is the equation to find the range for continuous variables?
range = URL for Xmax - LRL for Xmin
what is the equation to find the range for discontinuous variables?
range = Xmax - Xmin
what is s2
sample variance
SD?
standard deviation
what is the most commonly used and important measure of variability?
standard deviation and variance for a population
briefly explain what is measured by the standard deviation and what is measured by the variance
standard deviation measures the standard distance from the mean and variance measures the average squared distance from the mean
calculate the variance for the following population N=5 scores: 4, 0, 7, 1, 3
sum of scored deviations is 30, so the variance is 30/5= 6
what is ∑(X-μ)?
summation (score - population mean) = 0
the range can be defined as:
the difference between the upper real limit (Xmax) for the largest score, and lower real limit for the smallest score (Xmin)
standard deviation is:
the square root of the variance and provides a measure of the standard, or average, distance from the mean.
SS or Sum of Squares refers to:
the sum of the squared deviation scores
what is the standard deviation for the following set of N=5 scores: 10, 10, 10, 10, 10
there is no variability. the standard deviation is 0.