chapter 3

What is the relationship between the variance and the standard deviation?

Variance is the square of the standard deviation.

population variance

σ²=Σ(x-μ)²/N

sample mean of data in a frequency distribution

x̅= ΣfM / n

Which of the following accurately describes the median of a set of data?

The midpoint of the data when it is arranged in order from min to max

For which of the following variables can one calculate an arithmetic mean?

Time to run a marathon Daily temperatures in August for the past 10 years

In the formula for calculating the mean of grouped data, M stands for:

the midpoint of a given class

What does a small value for a measure of dispersion tell us about a set of data?

It indicates that the data is closely clustered around the center.

Of the following, which one is an advantage of the standard deviation over the variance?

It is in the same units as the data.

Which of the following statements is true for a mean or standard deviation calculated for grouped data?

It is only an estimate of the corresponding actual value.

Why would one use a grouped mean or standard deviation?

Only the frequency distribution data is available.

why would you use grouped mean or standard deviation

Only the frequency distribution data is available.

What characteristic of a data set makes the median the better measure of the center of the data than the mean?

When the data set includes one or two very large or very small values.

mean of grouped data

X( with line on top) =ΣfM/n

symmetrical distribution

all three measures of lotion are at the center of the distribution the mean median and mode have the same value

The variance is the mean of the sum of the squared deviations between each observation and the median.

false

major characteristics of the arithmetic mean

- at least the interval scale of measurement is required - all the data values are used in the calculation - a set of data has only one mean. that is, it is unique - the sum of the deviation between each observation and the mean is always 0 important properties - All of the values in the data are used in calculating the mean. - Σ(X-X)=0 i.e. the sum of the deviations is zero. - There is only one mean for a set of data.

Suppose the wait time at the emergency room follow a symmetrical, bell-shaped distribution with a mean of 90 minutes and a standard deviation of 10 minutes. What percentage of emergency room patients will wait between 1 hour and 2 hours?

99.7% Reason: The waiting time of the patients in the emergency room between 1 hour and 2 hours is correct if it lies within plus and minus three standard deviations of the mean. x( with line on top) − 3s=90−(3*10)=60 = 1 hour x( with linen top) +3s=90+(3*10)=120=2 hours Therefore, approximately 99.7% of the emergency room patients will wait between 1 hour and 2 hours.

Why is it important to consider all the measures of location in reporting statistics?

Each of the measures has advantages and disadvantages in representing the data.

Which statement best describes the difference between the formula for Population and Sample variance?

For the sample variance, dividing by n-1 corrects a tendency to underestimate population variance. The sample variance measures deviations from the sample mean, whereas the population variance uses the population mean.

A disadvantage of using an arithmetic mean to summarize a set of data is that _______.

it can be biased by one or two extremely small or large values

range

max-min

What is the purpose of a measure of location?

pinpoint the center of a distribution of data it is also referred to as average

standard deviation of grouped data

s=√Σf(M−x with line)^2/n−1

Variance

the arithmetic mean of the squared deviations from the mean

Which of the following statements are reasons to study the dispersion of data? Select all that apply.

- It allows us to compare the spread in two or more distributions. - A small value for dispersion indicates that the data is closely clustered around the center.

the weighted mean

- a convenient way to compute the arithmetic mean when there are several observation of the same value - It is used with data that has repeated values, such as a frequency distribution. - The denominator of the weighted mean is always the sum of the weights. - it is a special case of the arithmetic mean - the formula is x bar = ∑ (w • x) / ∑w

population mean definition

- the sum of all values in the population divided by the number of values in the population - the arithmetic mean of all of the values in the population

the mode

- the value of the observation that appears most frequently - we can determine the mode for all levels of data - advantage of no being affected by extremely high or low values - disadvantages are that there can be no mode or multiple modes

geometric mean

- useful for finding the average change of percentage, ratios, indexes, or growth rates over time - will always be less than or equal to the arithmetic mean (never more than) -examples of what it could find Average percentage annual yield for a portfolio of stocks. Average growth rate for four years of sales figures.

The Empirical rule states approximately what percentage of observations will be found within some deviation from the mean for a normal distribution. Match the percentage of observation to the range.

68% - plus or minus one standard deviation. plus or minus one standard deviation. 95% - plus or minus two standard deviations plus or minus two standard deviations 99.7% - plus or minus three standard deviations plus or minus three standard deviations

What does a measure of dispersion tell us about a set of data? and what should we study the dispersion data

It tells us about the spread of the data. - A small value for dispersion indicates that the data is closely clustered around the center. - It allows us to compare the spread in two or more distributions.

Why is it important to consider measures of dispersion as well as measures of location when reporting statistics?

Measures of location do not tell us about the spread or clustering of data

How does the formula for the sample mean differ from the formula for population mean?

The formulas are functionally the same, but 'n' (the sample size) is used instead of 'N' (the population size).

Which statement is true with regard to differences in the formula for the population and samples variances?

The sample variance measures deviations from the sample mean, whereas the population variance uses the population mean.

Chebyshev's Theorem says that for any set of observations, the proportion of values that lie within k standard deviations of the mean is:

at least 1-1/k^2, where k is any value greater than 1

sample variance ans sample standard deviation

s²=∑(x-x with line)²/n-1 n represents the sample size standard just is the square root of sample variance

in the calculation of the arithmetic mean for grouped data, which value is used to represent all the values in a particular class?

the class mid point

the median

the midpoint of the values after they have been arranged in rank order

When you calculate the sample mean, you divide the sum of the values in the sample by

the number of values in the sample.

What does s represent in the formula for the standard deviation of grouped data variance?

the sample of standard deviation

The arithmetic mean is the sum of the quantitative observations divided by the total number of observations.

true

Which measure of dispersion results in units that are different from the data?

variance

sample mean

x̅= ΣX / n

population mean

μ = ΣX / N

The mean, median, and mode are all the same for which type of distribution?

A symmetrical distribution.