1.2 Summarizing Data Using Frequency Distributions Flashcards by Cesar A. Contla

A frequency distribution

summarizes the values of a numerical variable into a few intervals

helpful when working with large data sets

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the types of frequency

Absolute frequency

Relative frequency

Cumulative relative frequency

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Absolute frequency

The actual number of observations in each interval

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Relative frequency

The absolute frequency divided by the total number of observations

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Cumulative relative frequency

The relative frequencies added up from the first interval to the current interval

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the steps to construct a frequency distribution

Sort the data in ascending order.
Calculate the range of the data.

Range = Maximum value − Minimum value

Determine the desired number of intervals, k
Determine the interval width.

Interval width = Range/k

Construct a table based on the minimum value, maximum value, the desired number of intervals (k), and the interval width.
Assign the observations into the respective intervals. Each observation will only fall in one interval since the intervals will not overlap.

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a contingency table

used to present the frequency distributions for multiple categorical variables simultaneously

the data in a contingency table can display absolute frequencies or relative frequencies

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joint frequencies

the individual cells in a contingency table that results from the crossing point of two variables (one from column and one from the row)

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marginal frequencies

last row and last column of the contingency table

shows the total per variable

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confusion matrix

uses the contingency table to evaluate the performance of a classification model.

constructed to evaluate the performance of the prediction model

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chi-square test of independence

test relationships between different variables in a contingency table

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how to preform chi-square test of independence

Use marginal frequencies in the contingency table to construct another table with expected values of the observations
Compare the expected values to the actual values to derive the chi-square test statistic
Compare the chi-square test statistic to the chi-square critical value for a given level of significance

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chi-square test of independence

Test statistic > Critical value:

conclusion and implication

conclusion: Reject the claim of independence

implication: There is a significant association between the categorical variables

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chi-square test of independence

Test statistic < Critical value:

conclusion and implication

conclusion: Do no reject the claim of independence

implication: There is no significant association between the categorical variables

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in a frequency distribution, the absolute frequency measure most likely:

A: represents the percentages of each unique value of the variable.

B: represents the actual number of observations counted for each unique value of the variable.

C: allows for comparisons between datasets with different numbers of total observations.

B: represents the actual number of observations counted for each unique value of the variable.

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A histogram

uses a chart to present the distribution of numerical data

non-overlapping intervals

from frequency distribution table

useful in presenting the frequency distribution of numerical data

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A frequency polygon

similar to a histogram

However, rather than using bars, a frequency polygon plots each interval’s midpoint on the x-axis and the absolute frequency on the y-axis

points are connected through line segments

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A cumulative frequency distribution chart

can also be used to illustrate the cumulative frequency distribution.

This shows how many observations lie below a certain value.

Most observations will lie on the steep slope

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bar chart

more appropriate when handling the frequency distribution of categorical data

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types of bar chart

pareto chart

A grouped bar chart (or clustered bar chart)

stacked bar chart

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A Pareto chart

sorts the categories by frequency in descending order along with a cumulative relative frequency line

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A grouped bar chart (or clustered bar chart)

may also be used to show joint frequencies when there are multiple categorical variables.

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stacked bar chart

instead of grouped (cluster) bar chart, pile them up one over the other

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tree-map

the frequency distribution of categorical data can be displayed using a tree-map

made up of different colored rectangles that have areas that represent the frequency of each category

provides a clear picture of the category that has the highest frequency.

–> However, it may become challenging to read when there are too many sub-categories

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A word cloud (a.k.a. tag cloud)

used to illustrate the frequency of textual data, which is a type of unstructured data. It allows analysts to quickly spot the most frequent terms in a report/article Words which appear more frequently have bigger sizes in the word cloud, and different colors may indicate different sentiments

A line chart

useful at visualizing ordered observations typically used to present the change in data over time One of the most common applications of a line chart in the finance industry is showing stock price trend over time A line chart may accommodate more than one set of data

A bubble line chart

can be used to add a third variable into a two-dimensional line chart

A scatter plot

describes the joint variation in two numerical variables shows the correlation between the variables at a particular point in time, which can be none, linear, or non-linear The degree of association is shown by the distance between the data points and the line of best fit.

A scatter plot matrix

can be used to visualize pairwise associations for more than two variables.

A heat map

used to visualize the frequency distribution of categorical data It enhances the presentation of a contingency table by introducing a color spectrum based on the frequency distribution

which type of chart should we use to explore or present a relationship between variables?

Scatter Plot Scatter Plot Matrix Heat Map

which type of chart should we use to explore or present a comparison among categories?

Bar Chart Tree-map Heat map

which type of chart should we use to explore or present a comparison over time?

Line Chart (two variables) Bubble Line chart (three variables)

which type of chart should we use to explore or present a distribution with numerical data?

Histogram Frequency polygon cumulative distribution chart

which type of chart should we use to explore or present a distribution with categorical data?

Bar Chart tree-map Heat Map

which type of chart should we use to explore or present a distribution with unstructured data?

Word cloud

A bar chart is similar to a histogram except it offers an alternative presentation of the same data is this true or false?

false

the most common statistical measures

Measures of central tendency

indicate where the data are centered.

The most common central tendency measures

arithmetic mean median mode weighted mean geometric mean

population

all possible observations extremely difficult to collect data on an entire population

sample statistic

subset of the population can then be used to draw inferences about the population statistic

the most common measure of where the data are centered

The arithmetic mean

the sum of the observations divided by the number of observations basically, the easiest type of average

The sample mean

the arithmetic mean for a sample

The value of the mean is extremely sensitive to extreme values or outliers true or false

true

three ways to deal with outliers

No adjustments: This is appropriate if all values are equally important and meaningful. Remove all outliers A trimmed mean Replace outliers with another value: A winsorized mean

a trimmed mean

removing all outliers calculated by discarding a certain percentage of the highest and lowest values For example, with a sample of 100 observations, a 2% trimmed mean would be the arithmetic mean without the highest value (top 1%) and the lowest value (bottom 1%).

a winsorized mean

adjusts any outliers' values to either an upper or lower limit. No observations are excluded from the calculation Replaces outliers with another value

The median

the middle item of a sorted list does not use all information about the observations because it only focuses on their relative position more complicated to calculate (less mathematically tractable) than the mean

if n is odd, what is the median

(n+1) / 2 th term

if n is even, what is the median

the mean between (n/2) and ((n+2)/2)

The mode

the most frequently occurring value in a distribution

The mode

the most frequently occurring value in a distribution

Some distributions have more than one mode, while others have none true or nah

true

A distribution with just one mode

unimodal

A distribution with two modes

bimodal

modal intervals

mode for data grouped in intervals it would be the interval with the highest bar

The weighted mean formula

X = Wn*n

uses of weighted average in finance

used to calculate past portfolio or index returns They can also be used to calculate future expected returns by weighting various scenarios

The geometric mean

used to average rates over time or compute growth rates It is often used to average portfolio returns from different time periods The geometric mean is always less than or equal to the arithmetic mean

formula for geometric mean

n root of the multiplication of (1 +return) for all the periods necessary -1 under the root

The harmonic mean

not as commonly used The observation's weight is inversely proportional to its magnitude Smaller weights are assigned to larger observations This property reduces the sensitivity of the harmonic mean to extremely large outliers

harmonic mean formula

1 / ((1/n)*(E * 1/Xi))

when is the harmonic mean useful

when the data consists of ratios (e.g., P/Es) It would also be appropriate if the analyst wants the average price paid for a security when investing the same dollar amount for several time periods (also known as the cost averaging technique)

is the harmonic mean always more or less than the geometric mean? what about the arithmetic mean?

always less always less

which Mean to use if we have a sample and we want to include all values including outliers?

arithmetic mean

which Mean to use if we have a sample and we want to do compounding?

geometric mean

the quantile

a value at or below which a stated fraction of the data is found If we arrange the observations in ascending order

most common quantiles

our quartiles five quintiles ten deciles one hundred percentiles

The y th percentile

the value at or below which y percent of the observations lie For example, the 90th percentile score (P90) on an exam is the number that separates the top 10% scores from the bottom 90%

The interquartile range (IQR)

the difference between the third quartile and the first quartile IQR = Q3 - Q1

how is the location of the y th percentile (Ly) in a list of n observations ranked?

ranked in ascending order (lowest to highest) is calculated as follows: Ly = (n + 1) * y/100

how do you find the percentile itself (Py)?

you do a weighted average depending on the decimals of the number equaling to Ly (location of percentile) if Ly = 14.07 you do: n14 * (1 - 0.07) + n15 *0.07 if Ly = 16.63 you do: n16* (1 - 0.63) + n17 *0.63

The dispersion of data across quartiles can be visualized how?

using a box and whisker plot

box and whisker plot

The "box" has a height equal to the interquartile range and is connected by two "whiskers." The two whiskers are bounded by the "fences," the fences are the highest and the lowest values of the observations

dispersion (or variability) around the mean

dispersion addresses risk

The most common measures of absolute dispersion

range mean absolute deviation variance standard deviation

The range

the difference between the maximum and minimum values limited as a dispersion measure because it uses only the highest and lowest values

The mean absolute deviation (MAD)

uses all the observations in the sample, which makes it better than the range

MAD formula

Multiplication of all (Mean of sample i - general mean) / n