Chapter Two

Histogram

Histogram

A bar graph wherein the bars touch each other; the heights of the bars correspond to the frequency values

What does the the horizontal scale represent on histograms?

Classes of quantitative data values

What does the the vertical scale represent on histograms?

Frequencies

Relative Frequency Histogram

A graph with the same shape and horizontal scale as a histogram but the vertical scale is marked with relative frequencies

Two types of misleading graphs

Nonzero axis and pictographs

Median

The middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

If the number of data values is odd, the median is ___

the number located in the exact middle of the list

If the number of data values is even, the median is ___

found by computing the mean of the two middle numbers

Percentiles (Pk)

A value in a set of data such that k percent of the observations are less than or equal to the value

Percentile of x (Px)=

number of values less than x/total number of values x 100

Percentiles are used to give the ___

relative standing of an observation

Quartiles

Divides the data set into fourths

Q1

Denotates the 25th percentile

Q2

Denotates the 50th percentile

Q3

Denotates the 75th percentile

The interquartile range (IQR)

The difference between Q3 and Q1; describes the variability of the middle 50% of the data values

5-number summary

A subset of the data that consists of the minimum value, the first quartile, the median, the third quartile, and the maximum value

Boxplots/box-and-whisker plots are constructed from ___

5-number summary

What do boxplots show?

How far extreme values are from the bulk of the data

Strengths of boxplots

Give a direct look at location and spread, identifies outliers, is great for comparing

Weaknesses of boxplots

Symmetry and skewness can be judged but shape cannot

Outlier

A data point that is not consistent with the bulk of the data from that group

Measure of center

A value at the center or middle of a data set such as the mean, median, mode, or midrange

Mode

The value that occurs with the greatest frequency in a data set

Bimodal

When two data values occur with the same frequency

Multimodal

When more than two values occur with the same frequency

Arithmetic Mean

The average value of a data set found by adding the data values and dividing the total by the number of data values

∑

The sum of a set of data values

x

The variable usually used to represent the individual data values

n

The number of data values in a sample (sample size)

N

The number of data values in a population (population size)

x̄= ∑x/n

The mean of a set of sample values (sample mean)

μ = ∑x/N

The mean of all values in a population (population mean)

Resistant

If extreme values (very large or very small) relative to the data do not affect its value substantially, then, the data is said to be resistant

Bell-shaped Distribution

The highest frequency occurs in the middle and frequencies tail off to the left and the right of the middle

Right Skewed

The tail extends to the right of the peak longer than to the left

Left Skewed

The tail extends to the left of the peak longer than to the right

Variation

the degree to which data are spread out

Range (R)

The difference between the maximum and minimum data value

Standard Deviation

A measure of the spread of the data

Population Standard Deviation equation

[See slideshow]

Sample Standard Deviation

[See slideshow]

The standard deviation is a measure of ___

how much data values deviate from the mean

The values of standard deviation can never be what?

Negative

If all the values are exactly the same, what is the standard deviation?

Zero

Larger values of standard deviation indicates what?

greater amounts of variation

What can drastically change the value of the standard deviation?

Outliers

Variance

Deviation about the mean; square of the standard deviation

Population variance

Sum of the squared deviations about the population mean divided by the number of observations in the population

Population variance equation

[See slideshow]

Sample variance equation

[See slideshow]

Sample variance

The sum of the squared deviations about the sample mean and divided by n-1

z Score

The number of standard deviations that a given value x is above or below the mean. The z-score is unitless, it has a mean of 0 and a standard deviation of 1.

Population z-score equation

[See slideshow]

Sample z-score equation

[See slideshow]