lecture four (more descriptive statistics) Flashcards
(22 cards)
What is the standard deviation?
Measures how spread out numbers are in a set.
Formula standard deviation.
Measures how spread out numbers are in a set.
It is a measure of dispersion that is in the same units as the data.
What does variance measure?
Variance measures how closely the observations are spread toward the mean.
It takes every data value into account.
Variance continue..
Low variance- the numbers are close and similar toward the mean
High variance- The numbers are very different and spread far from the mean.
Example: If students in a class all score around 80, 82, and 79, the variance is low (scores are close to the average).
If students score 50, 90, and 30, the variance is high (scores are very different from each other).
Formula for variance
for variance= s² = Σ (xi - x̄)² / (n - 1)
s² -> Sample variance (used when working with a sample, not full population)
Σ -> Sum of all values
xi -> Each sample value
x̄ -> Sample mean (average of sample values)
(xi - x̄)² -> Squaring the differences to remove negative values
n -> Sample size
(n - 1) -> Bessel’s correction (adjusts for small sample bias) to make the sampling more accurate.
Formula for population variance
for population variance= σ² = Σ (xi - μ)² / N
xi -> Each value in the dataset.
μ -> The mean (average) of all values.
(xi - μ) -> How far each value is from the mean.
(xi - μ)^2 -> Squaring the differences so they are always positive.
Σ -> Sum up all these squared differences.
N -> Divide by the total number of values (population size).
Define deviation in statistics.
The difference between a given observation and the mean of the sample is called its deviation.
What are the measures of variation?
- Standard deviation
- Quartile deviation
- Coefficient of variation
Formula for the coefficient of variation (Cv)?
Cv = (Standard deviation x 100) / mean
Formula for quartile deviation.
Quartile deviation = (Q3 – Q1) / 2
What does skewness measure?
Skewness measures the degree of asymmetry of a distribution relative to a normal distribution.
What indicates a positively skewed distribution?
In a positively skewed distribution: mean > median > mode.
What is kurtosis?
Kurtosis is a measure of whether data are peaked or flat relative to a normal distribution.
Formula to calculate the mean in a grouped frequency distribution?
Mean = Ʃf_i M_i / Ʃf_i
What does a higher coefficient of variation indicate?
A higher coefficient of variation indicates greater variability relative to the mean.
What defines a normal distribution?
A normal distribution is symmetrically bell-shaped.
What does the total of squared deviations represent in variance calculation?
It represents the total variation from the mean.
What is the definition of the arithmetic mean?
The arithmetic mean is calculated by the sum of observations divided by the total number of observations.
What is the significance of the sample size in variance calculations?
The denominator has 1 subtracted from it to adjust for sample size underestimating the true variation.
What is the quartile deviation for the given example?
Quartile deviation = (Q3 – Q1) / 2
What does a positive kurtosis indicate?
A positive kurtosis indicates a relatively peaked distribution.
What is the skewness formula according to Pearson?
SK = 3(mean – median) / standard deviation
