Mathematics in the Modern world

spread/variability/dispersion

Measures of variation give information on the ____/____/____ the data values.

Coefficient of Variation

Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare the variability of two or more sets of data measured in different units

Skewness

Measures the amount of asymmetry in a distribution

Kurtosis

Measures the relative concentration of values in the center of a distribution as compared with the tails

coefficient of correlation

Measures the relative strength of the linear relationship between two numerical variables

Geometric mean rate of return

Measures the status of an investment over time

right-skewed

Median < Mean

Population standard deviation

Most commonly used measure of variation Shows variation about the mean Is the square root of the population variance Has the same units as the original data

Sample standard deviation

Most commonly used measure of variation n Shows variation about the mean n Is the square root of the variance n Has the same units as the original data

T

A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0. T/F?

99.7%

Approximately _____ of the data in a bell-shaped distribution lies within three standard deviations of the mean, or µ ± 3σ

95%

Approximately _____ of the data in a bell-shaped distribution lies within two standard deviations of the mean, or µ ± 2σ

68%

Approximately ______ of the data in a bell shaped distribution is within 1 standard deviation of the mean or µ ±1 σ

Sample variance

Average (approximately) of squared deviations of values from the mean

Arithmetic mean Median Mode Geometric Mean

Central Tendency covers 4 measurements including

objective

Data analysis is ________ Should report the summary measures that best describe and communicate the important aspects of the data set

subjective

Data interpretation is ______ Should be done in fair, neutral and clear manner

T

Descriptive statistics discussed a sample, not the population. T/F?

Q1= (n+1)/4

First quartile position

even

If the number of values is ______ , the median is the average of the two middle numbers

odd

If the number of values is _______ , the median is the middle number

zero

If the values are all the same (no variation), all these measures will be ______.

median

In an ordered array, the _______ is the "middle" number and is not affected by extreme values

The Interquartile Range

It is Q3 - Q1 and measures the spread in the middle 50% of the data It is also called the midspread because it covers the middle 50% of the data It is a measure of variability that is not influenced by outliers or extreme values

left-skewed

Mean < Median

Symmetric

Mean = Median

F, variation

None of measures of central tendency are ever negative. T/F? if F, what is the correct answer?

Parameters

Population is for ______

Chebyshev Rule

Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)

statistics

Sample is for ______

Q2= (n+1)/2

Second quartile position

Range

Simplest measure of variation Difference between the largest and the smallest values

mean

Sum of values divided by the number of values Affected by extreme values (outliers)

shape

The ______ of distribution describes how data are distributed

Z-score

The _______ is the number of standard deviations a data value is from the mean.

(n+1)/2

The location of the median when the values are in numerical order (smallest to largest) can be determine if this formula is used.

smaller

The more the data are concentrated, the _______ the range, variance, and standard deviation.

greater

The more the data are spread out, the _____ the range, variance, and standard deviation.

F, larger

The smaller the absolute value of the Z-score, the farther the data value is from the mean. T/F?

Q3= 3(n+1)/4

Third quartile position

Geometric mean

Used to measure the rate of change of a variable over time

Mode

Value that occurs most often Not affected by extreme values Used for either numerical or categorical (nominal) data There may may be nothing There may be several

Ignores the way in which data are distributed Sensitive to outliers

Why The Range Can Be Misleading?

variation

amount of dispersion or scattering of values

opposite

cov(X,Y) < 0: X and Y tend to move in ______ directions

independent

cov(X,Y) = 0 X and Y are ___________

same

cov(X,Y) > 0: X and Y tend to move in the _____ direction

central tendency

extent to which all the data values group around a typical or central value.

mean

generally used, unless extreme values (outliers) exist

covariance

measures the strength of the linear relationship between two numerical variables (X & Y) Only concerned with the strength of the relationship No causal effect is implied Major flow: It is not possible to determine the relative strength of the relationship from the size of the covariance

arithmetic mean/ mean

most common measure of central tendency

median

often used, since it is not sensitive to extreme values. For example, ____ home prices may be reported for a region; it is less sensitive to outliers.

shape

pattern of the distribution of values from the lowest value to the highest value.

sum of the values in the population divided by the population size

population mean

population mean

sum of the values in the population divided by the population size