Organizing and Summarizing Data – Numerical Statistics

Organizing and Summarizing Data Chapters 2 & 3

In chapters 2 and 3, we look at organizing and summarizing data in numerical statistics.


raw data

data obtained from either observational studies or designed experiments, before it is organized into a meaningful form.


frequency distribution

lists each category of data and the number of occurrences for each category of data


relative frequency

the proportion (or percent) of observations within a category


relative frequency equation

relative frequency = frequency / sum of all frequencies


relative frequency distribution

Lists each category of data together with the relative frequency. The sum of all the relative frequencies should add up to 1.


bar graph

Constructed by labeling each category of data on either the horizontal or vertical axis and the frequency or relative frequency of the category on the other axis. Rectangles of equal width are drawn for each category. The height of each rectangle represents the category’s frequency or relative frequency.


Pareto chart

a bar graph whose bars are drawn in decreasing order of frequency or relative frequency


side-by-side bar graph

Compares two sets of data by aligning the bars for one data set with the bars for another data set, by class. Should be compared using relative frequencies to avoid differences in population sizes.


pie chart

A circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category.


histogram

Constructed by drawing rectangles for each class of data. The height of each rectangle is the frequency or relative frequency of the class. The width of each rectangle is the same and the rectangles touch each other.


lower class limit

the smallest value within a class


upper class limit

the largest value within a class


class width

the difference between consecutive lower class limits


open ended

a class whose first class has no lower class limit, or whose last class has no upper limit


guidelines for determining the lower class limit of the first class and class width

1. choose the lower class limit of the first class by choosing the smallest observation in the data set or a number slightly lower than the smallest observation in the data set
2. determine the class width by deciding on the number of classes, then compute and round up:
class width = (largest data value – smallest data value)/number of classes


stem-and-leaf plot

a method of representing quantitative data graphically by using the digits to the left of the rightmost digit to for the stem, and the rightmost digits to form the leaf


dot plot

graph drawn by placing each observation horizontally in increasing order and placing a dot above the observation each time it is observed


distribution shapes

1. uniform distribution
2. bell-shaped distribution
3. skewed right
4. skewed left


uniform distribution

the frequency of each value of the variable is evenly spread out across the values of a variable


bell-shaped distribution

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


skewed right

the tail to the right of the peak is longer than the tail to the left of the peak


skewed left

the tail to the left of the peak is longer than the tail to the right of the peak


time-series data

data collected on the same element for the same variable at different points in time or for different time periods


time-series plot

Obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Line segments are then drawn connecting the points.


misleading graph

a graph that unintentionally creates an incorrect impression


deceptive graph

a graph that purposely attempts to create an incorrect impression


graphical misrepresentations of data

1. misrepresentation of data
2. misrepresentation of data by manipulating the vertical scale
3. misleading graphs


guidelines for constructing good graphics

1. Title and label the graphic axes clearly. Include units of measurement and a data source.
2. Avoid distortion. Never lie about the data.
3. Minimize the white space. Clearly indicate truncated scales.
4. Avoid clutter.
5. Avoid three dimensional graphs.
6. Do not use more than one design in the same graphic.
7. Avoid relative graphs that are devoid of data or scales.


arithmetic mean

computed by determining the sum of all the values of the variable in the data set and dividing by the number of observations


population arithmetic mean

a parameter computed using all the individuals in a population


sample arithmetic mean

statistic computed using sample data


μ (mu)

population arithmetic mean


x̄ (x-bar)

sample arithmetic mean


∑ (sigma)

the terms to be added, the sum


population mean equation

where x₁,x₂,…,x-sub-n are the N observations of a variable from a population:
μ =(x₁+x₂+…+x-sub-n)/n = (∑x-sub-i)/n


sample mean

where x₁,x₂,…,x-sub-n are the N observations of a variable from a sample:
x̄ =(x₁+x₂+…+x-sub-n)/n = (∑x-sub-i)/n


median

The value that lies in the middle of the data when arranged in ascending order. If the number of observations is even, the median is the average of the two middle observations.


M

median


resistant numerical summary

extreme values relative to the data do not affect its value substantially. median is resistant, mean is not.


skewed left distribution shape

mean is substantially smaller than median


skewed right distribution shape

mean is substantially larger than median


symmetric distribution shape

mean is roughly equal to median


mode

the most frequent observation of the variable that occurs in the data set


no mode

no observation occurs more than once


bimodal

two observations within a data set occur with equal frequency


multimodal

three or more observations within a data set occur with equal frequency


dispersion

the degree to which the data are spread out


range

the difference between the largest data value and the smallest data value


R

range


deviation about the mean

How far, on average, each observation is from the mean.


sum of all deviations about the mean

must equal zero.
∑(x-sub-i – µ) = 0 and ∑(x-sub-i – x̄) = 0


σ

population standard deviation


population variance

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


σ²

population variance


population variance equation

σ² = [(x₁-µ)² + (x₂-µ)² + (x-sub-n-µ)² ]/N

or
σ² = [∑(x-sub-i-µ)²]/N


sample variance

computed by determining the sum of the squared deviations about the sample mean and dividing this result by n-1


sample variance


sample variance equation

s² = [∑ (x-sub-i – x̄)²]/(n-1)


biased

a statistic that consistently overestimates or underestimates a parameter


degrees of freedom

n-1 observations can be whatever they want, but the nth value must be whatever value forces the sum of the deviations about the mean to equal zero


population standard deviation

σ = √σ²


sample standard deviation

s = √s²


empirical rule

if a distribution is bell shaped:
68% of the data will lie within 1 standard deviation
95% of the data will lie within 2 standard deviations
99.7% of the data will lie within 3 standard deviations


Chebyshev’s inequality

For any data set, regardless of the shape of the distribution, at least [1-(1/k²)]100% of the observations will lie within k standard deviations of the mean, where k is greater than 1.


class midpoint

found by adding consecutive lower class limits and dividing the result by 2


population mean

µ = ∑xifi /∑fi


steps to approximate the mean

1. Determine the class midpoint of each class
2. Compute the sum of the frequencies
3. Multiply the class midpoint by the frequency to obtain xifi for each class
4. Compute ∑xifi
5 Calculate the mean using x̄ = ∑xifi /∑fi


sample mean

x̄ = ∑xifi /∑fi


weighted mean

mean calculated when certain data values have a higher importance or weight associated with them


weighted mean equation

x̄w = ∑wixi/∑wi

multiply each value of the variable by its corresponding weight, sum the products, and divide the result by the sum of the weights


approximate population variance from a frequency distribution

σ² = [ ∑(xi – µ)² fi]/∑fi

where xi is the midpoint or value of the ith class
fi is the frequency of the ith class


approximate sample variance from a frequency distribution

s² = [∑(xi – x̄)² fi] / ∑fi – 1

where xi is the midpoint or value of the ith class
fi is the frequency of the ith class


z-score

The distance that a data value is from the mean in terms of the number of standard deviations. Obtained by subtracting the mean from the data value and dividing this result by the standard deviation. Unitless, mean is 0, standard deviation is 1.


population z-score

z = (x – µ)/σ


sample z-score

z = (x – x̄)/s


kth percentile

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


Pk

kth percentile


quartiles

Percentile that divides the data into fourths.
First quartile = 25th percentile
Second quartile = 50th percentile
Third quartile = 75th percentile


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. Q₁ is the median of the bottom half, Q₃ is the median of the top half.


interquartile range

the range of the middle 50% of the observations in a data set; the difference between the first and third quartiles


interquartile range equation

IQR = Q₃ – Q₁


describe the distribution

1.describe the shape (skewed left, skewed right, or symmetric)
2. describe the center (mean or median)
3. describe the spread (standard deviation or interquartile range)


outliers

extreme observations in the data set


check for outliers using quartiles

1. Determine the first and third quartiles of the data
2. Compute the interquartile range
3. Determine the lower and upper fences
4. If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier


fences

cutoff points for determining outliers
Lower fence = Q₁ – 1.5(IQR)
Upper fence = Q₃ + 1.5(IQR)


exploratory data analysis

examination of data in order to describe their main features using statistical tools and ideas


five-number summary

MINIMUM Q₁ M Q₃ MAXIMUM


Constructing a boxplot

1. Determine the lower and upper fences
2. Draw vertical lines at Q1, M, and Q3. Enclose those vertical lines in a box.
3. Label the lower and upper fences.
4a. Draw a line (whisker) from Q1 to the smallest data value that is larger than the lower fence.
4b. Draw a line (whisker) from Q3 to the largest data value that is smaller than the upper fence.
5. Mark any data value less than the lower fence 0r greater than the upper fence (outliers) with an asterisk.


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