MATH 120 Quiz#6 (Sections 3.3-3.5)

Quartiles

1st quartile (Q1): divides the bottom 25% of the data from the top 75%. Equivalent to the 25th percentile. 2nd quartile (Q2): divides the bottom 50% of the data from the top 50%. Equivalent to the median. 3rd quartile (Q3): divides the bottom 75% of the data from the top 25%. Equivalent to the 75th percentile.

kth Percentile

A value such that k percent of the observations are less than or equal to the value.

Five-Number Summary (3.5)

Consists of the smallest data value Q1, the median, and the largest data value Q3.

Interquartile Range (IQR)

The range of the middle 50% of the observations in a data set. Is similar to that of the range and standard deviation whereas the more spread a data set has, the higher the interquartile range will be. IQR = Q3 -Q1

Sample Standard Deviation

s = √∑(x1 - μ)^2 f2/(∑f1) - 1 x1 = midpoint or value of the ith class f1 = frequency of the ith class

Sample Mean

x̄ = ∑ xifi / ∑ fi = x1f1 + x2f2 + ... + xnfn / f1 + f2 + ... + fn f1 = frequency of the ith class n = number of classes

Sample Z-Score

z = x - x̄/s

Population Z-Score

z = x - μ/σ

Population Mean

μ = ∑ xifi / ∑ fi = x1f1 + x2f2 + ... + xnfn / f1 + f2 + ... + fn f1 = frequency of the ith class n = number of classes

Population Standard Deviation

σ = √∑(x1 - μ)^2 f2/∑f1 x1 = midpoint or value of the ith class f1 = frequency of the ith class

Outliers

Extreme values that don't appear to belong with the rest of the data.

Finding Quartiles

1) Arrange the data in ascending order. 2) Determine the median, M, or second quartile, Q2. 3) Determine the first and third quartiles, Q1 and Q3, by dividing the data set into two halves; the bottom half will be the observations below (to the left of) the location of the median. The first quartile is the median of the bottom half and the third quartile is the median of the top half.

Checking for Outliers by Using Quartiles

1) Determine the first and third quartiles of the data. 2) Compute the interquartile range. 3) Determine the fences. Fences serve as cutoff points for determining outliers. Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) 4) If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier.

Drawing a Boxplot

1) Determine the lower and upper fences. 2) Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q 1 , M, and Q 3. Enclose these vertical lines in a box. 3) Label the lower and upper fences. 4) Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. These lines are called whiskers. 5) Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).

Grouped Data (3.3)

Data that has been summarized in a frequency distribution.

Z-Score (3.4)

The distance that a data value is from the mean in terms of the number of standard deviations. Is unitless and has a mean of 0 and a standard deviation of 1.

Weighted Mean

The mean found by multiplying each value by its corresponding weight and dividing by the sum of the weights.