3-2 Measurements of Variation
__________________ the highest value minus the lowest value. The symbol R is used for the range
R =maximum (highest) value - minimum (lowest) value
An outlier (an extremely high or low value) can affect the range.
_____________ variance is the difference (or distance) from the mean. X - 𝜇 (Observation - mean) This
difference is called the deviation.
If we sum up the deviations for all the observations of the data set, it would be zero. So we sum the squares
𝛴 = (𝑋 − 𝜇)
2
. We then divide by N, the number of observations in the population(𝑋 −
𝜇) 2 /𝑁
.This would be the population variance. We divide by N, the number of observation if we
have the total population values.
____________________ is the average of the squares of the distance that each value is from the mean.
_________________________ _________________________ is the ______________ ___________
of the variance
To find the standard deviation, we take the square root of the variance,
√(𝑋 − 𝜇)
2 /𝑁
Uses of the Variance and Standard Deviation
-To determine the spread of the data.
-To determine the consistency of a variable.
-To determine the number of data values that fall within a specified interval in a distribution (Chebyshev’s
Theorem).
-Used in inferential statistics
Finding the Population Variance and Population Standard Deviation
Step 1 Find the mean for the data.
Step 2 Find the Deviation for each data value.
Step 3 Square each of the deviations.
Step 4 Find the sum of the squares.
Step 5. Divide by N, to get the variance
(𝑋 − 𝜇)
2
/𝑁
√
Step 6. Take the square root of the variance to get the standard deviation
If the observations are near the mean, the variance will be small.
If the observations are far from the mean, the variance will be large.
If we are using a sample from a population, we use N - 1 to divide.
(𝑋 − 𝜇)
2
/𝑁