Measures of Central Tendency
What are the three main measures of central tendency?
Mean, median, and mode
How do you determine the median in an odd-numbered data set?
It is the middle value when numbers are arranged in order
What are the three main measures of variability?
Range, variance, and standard deviation
What symbol represents the population mean?
The Greek letter μ (mu)
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance
What is the purpose of central tendency in statistics?
It helps compare individual scores to a distribution of scores
In a normal distribution, where do mean, median, and mode fall?
At the same spot; they are all equal to each other
Why is the range not a robust measure of spread?
It is highly sensitive to outliers
What is the formula for sample variance?
Sum of squares (SS) divided by degrees of freedom (N-1)
Why is standard deviation more commonly used than variance?
It is in the same units as the original data, making it easier to interpret
What measure of central tendency is used for qualitative or categorical data and why?
The mode; because we are looking for a category to be the most common answer when collecting qualitative/categorical data
What is the key disadvantage of using the mean?
It is sensitive to extreme values (outliers) and skewness
What is sum of squares?
By using the mean as the measure of the middle of the distribution, we can see how far, on average, each data point is from the center (how widely a dataset is spread from the mean)
If the sum of squares (SS) is 42 and the N is 6, what is the variance?
42/(6-1) = 8.4
What does a larger standard deviation indicate?
More variability in the data
What's the difference between the population mean equation and sample mean equation?
pop: mu = (Simga(x))/N
sample: x-bar/hat = (Sigma(x))/n
When is it best to report the median instead of the mean?
When the data set is skewed or contains extreme outliers
What does variance measure in a data set?
The average squared deviation from the mean
If the variance is 7.75, how do you find the standard deviation?
Take the square root of 7.75
If one distribution has a mean of 40 and an SD of 5, and another has a mean of 60 and an SD of 10, which is more spread out?
The second distribution (higher SD means more spread)
Why is central tendency fundamental to statistics?
It provides a way to summarize data sets with a single representative value
In a negatively skewed distribution, where do mean, median, and mode fall?
Mode is the peak, median is to the left of mode, and mean is to the left of median
What is the long version of this standard deviation equation: sqrt(SS/df)?
s = sqrt((Sigma(x-x-hat)^2/(N-1))
What is the sum of squares (SS) for this set of data: x1 = 5, x2 = 7, x3 = 9, x-hat = 7?
8
Why do we rarely use population standard deviation in practice?
We almost never have access to an entire population’s data