Quiz 2: Central Tendency and Variability

What is the statistical definition of the 'mode'?

The most frequently occurring value in a distribution.

What is the standard abbreviation for the mode?

$Mo$

A distribution with exactly two modes is described as __.

Bimodal

Which measure of central tendency is the only one appropriate for nominal-level variables?

The mode

In statistics, what does the 'median' represent?

The value of the middle case in an ordered distribution.

What is the standard abbreviation for the median?

$Mdn$

The median is an appropriate measure of central tendency for data at which levels of measurement?

Ordinal or interval levels.

When calculating the median, what must be done to the cases before identifying the middle value?

The cases must be ordered (e.g., lowest to highest).

How is the median determined if the total number of cases ($N$) in a distribution is odd?

Choose the middle-most case as the median.

How is the median determined if the total number of cases ($N$) in a distribution is even?

Interpolate between the two middle cases.

Formula: What is the equation to find the position of the median?

Position of the $Mdn = \frac{N+1}{2}$

Which measure of central tendency is defined as the mathematical average of a set of scores?

The mean

The mean is considered appropriate for variables at which two levels of measurement?

Interval and ratio levels.

Formula: What is the definitional equation for the mean ($\bar{X}$)?

$\bar{X} = \frac{\sum X}{N}$

In the formula $\bar{X} = \frac{\sum X}{N}$, what does the symbol $\sum$ represent?

The sum (of the scores).

In statistical notation for the mean, what does the symbol $N$ represent?

The total number of scores in a set.

What is a 'weighted mean'?

The overall mean calculated from subgroup means rather than raw data.

Formula: What is the equation for the weighted mean ($\bar{X}w$)?

$\bar{X}w = \frac{\sum N{group} \bar{X}_{group}}{N{total}}$

In a symmetrical distribution, what is the relationship between the values of the mode, median, and mean?

They have identical values.

Which measure of central tendency is most heavily influenced by extreme outliers in a skewed distribution?

The mean

Why is the median often preferred over the mean in highly skewed distributions?

It is not influenced by extreme outliers.

According to the 'Applicability of Central Tendency Measures' table, which measures can be used for Ordinal data?

Mode and Median.

Why are measures of central tendency considered incomplete on their own?

They do not show how scores are distributed around the center.

What is the general purpose of measures of variability?

To index how scores are spread or distributed around the center of a distribution.

Term: Range

The difference between the highest and lowest scores in a distribution.

Formula: What is the equation for calculating the Range ($R$)?

$R = H - L$

What is the primary disadvantage of using the range as a measure of variation?

It is a crude measure easily affected by outliers.

Which measure of variability is calculated as the difference between the 3rd quartile and the 1st quartile?

The Inter-Quartile Range ($IQR$)

Formula: What is the equation for the Inter-Quartile Range ($IQR$)?

$IQR = Q3 - Q1$

What is the specific advantage of the Inter-Quartile Range over the standard range?

It manages the effects of extreme outliers.

Term: Mean Deviation ($MD$)

The mean of the absolute differences between individual scores and the mean of the distribution.

Formula: What is the equation for the Mean Deviation ($MD$)?

$MD = \frac{\sum |X - \bar{X}|}{N}$

In statistics, what is a 'deviation'?

The distance of any given raw score from its mean ($X - \bar{X}$).

When calculating variability, why are deviations from the mean squared before being summed?

To ensure the sum of deviations is non-negative (greater than or equal to 0).

Term: Variance ($s^2$)

The average of the squared deviations from the mean.

Formula: What is the definitional formula for Variance ($s^2$)?

$s^2 = \frac{\sum (X - \bar{X})^2}{N}$

What is the primary interpretative difficulty when using variance as a measure of variability?

The unit of measurement is squared.

How is the standard deviation ($s$) derived from the variance ($s^2$)?

By taking the square root of the variance.

Formula: What is the definitional formula for the Standard Deviation ($s$)?

$s = \sqrt{\frac{\sum (X - \bar{X})^2}{N}}$

Variance and standard deviation apply only to variables at which level of measurement?

Interval level (and ratio level).

Formula: What is the computational (raw-score) formula for Variance ($s^2$)?

$s^2 = \frac{\sum X^2}{N} - \bar{X}^2$

Formula: What is the computational (raw-score) formula for Standard Deviation ($s$)?

$s = \sqrt{\frac{\sum X^2}{N} - \bar{X}^2}$

What does the standard deviation represent in terms of a distribution?

The average variability or average of deviations from the mean.

How does the size of the standard deviation relate to the variability of a distribution?

Greater variability results in a larger standard deviation.

Term: Coefficient of Variation ($CV$)

A measure of variation calculated as the standard deviation divided by the mean, then multiplied by 100.

Formula: What is the equation for the Coefficient of Variation ($CV$)?

$CV = 100 \cdot (\frac{s}{\bar{X}})$

When is it most appropriate to use the Coefficient of Variation ($CV$) instead of the standard deviation?

When comparing the variability of two distributions with different measurement units.

What is the goal of the 'raw-score' or 'computational' formulas for variance and standard deviation?

To provide an easier way to calculate these values using raw scores instead of deviations.

The sum of raw scores divided by $N$ equals the __.

Mean ($\bar{X}$)

In a simple frequency distribution table for variability, what does the column labeled '$fX^2$' represent?

The frequency of a score multiplied by the square of that score.

True or False: Skewness is considered relevant only at the interval level of measurement.

TRUE

Which measure of central tendency is described as the most stable in large data sets?

The mean