Chapter 3 Numerically Summarizing Data

Question 1

Measures of Central Tendency

Answer 1

Give a feel for where the center of gravity of the data set is: Mean, Median, Mode

Question 2

Arithmetic Mean

Answer 2

The average

Question 3

Median

Answer 3

the value that lies in the middle of the data set when arranged in ascending order

Question 4

Mode

Answer 4

The most frequent observation of the variable that occurs a the data set

A data set can have no mode, one mode, or more than one mode

Question 5

The symbol for the mean of a population

Answer 5

The Greek letter mu (μ)

Question 6

The symbol for the mean of a sample

Answer 6

(X ̅) “X-Hat”

Question 7

The symbol for the median of a population

Answer 7

M

Question 8

The symbol for the median of a sample

Answer 8

m

Question 9

Relation between the mean, median, and a distribution shape that is skewed to the left

Answer 9

Mean is substantially smaller than the median

Question 10

Relation between the mean, median, and a distribution shape that is symmetric

Answer 10

Mean roughly equal to median

Question 11

Relation between the mean, median, and a distribution shape that is skewed to the right

Answer 11

Mean substantially larger than median

Question 12

What does it mean when it is said that a data set is resistant?

Answer 12

Extreme values (very large or small) relative to the data do not affect its value substantially

Question 13

Outlier

Answer 13

A data point that differs significantly from other observations. Results in a skewed distribution

Question 14

What is the better measure of central tendency when the distribution is skewed?

Answer 14

The median

Question 15

Measures of Dispersion

Answer 15

Show the degree to which the data in a population or sample is spread out: range, standard deviation, variance, interquartile range

Question 16

Range

Answer 16

The difference between the largest data value and the smallest data value in a data set. Denoted as R

Range is not resistant to outlier values

Question 17

Population Variance

Answer 17

The sum of the squared deviations from the population mean divided by the number of observations in the population, N. Denoted by the greek letter sigma squared (σ^2)

Question 18

Population Standard Deviation

Answer 18

The positive square root of the population variance. Denoted by the Greek letter sigma (σ).

Question 19

The population variance and standard deviation are

Answer 19

Parameters

Question 20

Sample Variance

Answer 20

The sum of the squared deviations from the population mean divided by the size of the sample MINUS 1 (n - 1). Denoted by s squared (s^2).

Question 21

Sample Standard Deviation

Answer 21

The positive square root of the sample variance. Denoted by s

Question 22

The sample variance and standard deviation are

Answer 22

Statistics

Question 23

When calculating the sample variance, the denominator is

Answer 23

n - 1

Question 24

The Empirical Rule

Answer 24

States that, if the summary measures of mean (μ) and standard deviation (σ) are known, and if the distribution is approximately bell-shaped:
≈ 68% of the data will lie within ±1σ of the mean
≈ 95% of the data will lie within ±2σ of the mean
≈ 99.7% of the data will lie within ±3σ of the mean

Question 25

Outlier for a bell-shaped distribution

Answer 25

Any data point less than -3 standard deviations (3σ) from the mean or more than 3σ from the mean

Question 26

Z-Score

Answer 26

Represents the distance that a data value is from the mean in terms of the number of standard deviations. We find it by subtracting the mean from the data value and dividing this result by the standard deviation. Round z-scores to the nearest hundredth.

Question 27

Z-Score for a data value in a population

Answer 27

= (x-μ)/σ, where μ is the population mean and σ is the | population standard deviation

Question 28

Z-Score for a data value in a sample

Answer 28

z= (x-X ̅)/s, where X ̅ is the sample mean and s is the sample standard deviation

Question 29

kth percentile

Answer 29

A value such that at least k percent of the observations are less than or equal to this value, and at least (100-k) percent of the observations are greater than or equal to this value. Denoted Pk

Question 30

1st quartile

Answer 30

Denoted Q1, divides the bottom 25% of the data from the top 75%. Therefore, the 1st quartile is equivalent to the 25th percentile.

Question 31

2nd quartile

Answer 31

Denoted Q2, divides the bottom 50% of the data from the top 50% of the data, so that the 2nd quartile is equivalent to the 50th percentile, which is equivalent to the median.

Question 32

3rd quartile

Answer 32

Denoted Q3, divides the bottom 75% of the data from the top 25% of the data, so that the 3rd quartile is equivalent to the 75th percentile.

Question 33

Method for determining quartiles using Excel

Answer 33

To find Q1: =QUARTILE.EXC (highlight the data, 1) To find Q2 (The Median, M): =QUARTILE.EXC (highlight the data, 2) To find Q3: =QUARTILE.EXC (highlight the data, 3)

Question 34

Method for determining quartiles by Inspection

Answer 34

If the data set is relatively small, the direct “by inspection” method can be used: Step 1: Arrange the data in ascending order. Step 2: Determine the median, M, or second quartile, Q2 . Step 3: Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q1 , is the median of the bottom half, and the third quartile, Q3, is the median of the top half.

Question 35

Interquartile range

Answer 35

Defines the range of the middle 50% of the observations in a data set. Denoted as IQR = Q3 – Q1

Question 36

When is it best to use the median as the measure of central tendency and the interquartile range as the measure of dispersion and why?

Answer 36

When the distribution of data is highly skewed or contains extreme observations; because these measures are resistant.

Question 37

Method for checking a data set for outliers

Answer 37

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

Question 38

Five-number summary

Answer 38

Consists of the smallest data value, Q1, the median, Q3, and the largest data value. Used to learn information about the extremes of the data set.

Question 39

Method for constructing a box plot using the TI-84 calculator

Answer 39

Step 1: Type data into L1 Step 2: 2nd > STAT PLOT Step 3: Select PLOT 1 and set it to ON Step 4: Select the box plot with outliers graph (4th from left) Step 5: Press GRAPH button Step 6: If graph is not visible: ZOOM > 9: ZOOM STAT

Question 40

Resistant

Answer 40

A numerical summary of data is said to be resistant if extreme observations (very large or small) relative to the data do not affect its value substantially.

Question 41

Multimodal

Answer 41

Describes a data set that has three or more values that occur with the highest frequency

Question 42

Bias

Answer 42

Occurs whenever a statistic consistently underestimates or overestimates a parameter

Question 43

Degrees of freedom

Answer 43

For the sample standard deviation, we call n−1 the degrees of freedom because the first n−1 observations have freedom to be whatever value they wish, but the nth observation has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero.

Question 44

Describe the Distribution

Answer 44

Means to describe its shape (skewed left, skewed right, or symmetric), its center (mean or median), and its spread (standard deviation or interquartile range).

Question 45

Boxplot

Answer 45

A graphical summary of quantitative data used to identify the shape of a distribution and outliers.

Question 46

Bimodal

Answer 46

Describes a data set that has two values that occur with the highest frequency

Question 47

Deviation about the mean

Answer 47

For the ith observation in a population: xi – μ. | For the ith observation in a sample: xi − x ̅

Question 48

No mode

Answer 48

Occurs If no observation occurs more than once in a data set

Question 49

Quartiles

Answer 49

Divide data sets into fourths, or four equal parts.

Question 50

Measures of Position

Answer 50

z-sore, percentiles, outliers

Question 51

Method for Checking for Outliers by Using Quartiles

Answer 51

Step 1. Determine the first and third quartiles of the data. Step 2: Compute the interquartile range. Step 3: Determine the fences: LF = Q1 - 1.5 (IQR); UF = Q3 + 1.5 (IQR) Step 4: If a data point value is less than the lower fence or greater than the upper fence, it is considered an outlier.