AP Statistics 1.5-1.6: Representing Quantitative Data

What five values are needed to construct a standard box plot?

The five-number summary consists of the minimum, first quartile $Q_1$, median $Q_2$, third quartile $Q_3$, and maximum.

Why must a box plot use a properly scaled number line?

A scaled number line preserves the numerical distances among the five summary values. Without a scale, the visual lengths of the box and whiskers can be misleading.

How is the central box of a box plot interpreted?

The box extends from $Q_1$ to $Q_3$ and contains approximately the middle 50% of the data. Its length represents the interquartile range.

What do the whiskers represent in a standard box plot?

The whiskers extend from $Q_1$ to the minimum and from $Q_3$ to the maximum. Thus, they show the spread of the lower and upper portions of the data.

Interquartile range (IQR)

The interquartile range measures the spread of the middle 50% of a data set: $IQR=Q_3-Q_1$.

What does each quarter of a box plot represent?

Each interval from the minimum to $Q_1$, $Q_1$ to the median, the median to $Q_3$, and $Q_3$ to the maximum contains approximately 25% of the observations. The intervals can have different numerical widths.

What does it mean if one quarter of a box plot is much shorter than another?

The shorter interval has observations packed into a smaller numerical range, while the longer interval has greater spread. The number of observations in each quarter is still approximately the same.

Range

The range measures the overall spread of a data set: $\text{range}=\text{maximum}-\text{minimum}$.

For a data set with minimum $59$, $Q_1=64.5$, median $66$, $Q_3=70$, and maximum $77$, what are the range and IQR?

The range is $77-59=18$. The IQR is $70-64.5=5.5$.

For the five-number summary $59$, $64.5$, $66$, $70$, $77$, which quarter has the greatest spread?

The upper quarter, from $Q_3=70$ to the maximum $77$, has spread $7$, which is larger than the other quarter spreads.

How can two data sets be compared with box plots?

Use a common scaled number line and compare their centers, overall ranges, IQRs, and the lengths of corresponding sections. A larger IQR indicates greater variability in the middle 50%.

If one data set has a larger IQR than another, what conclusion is justified?

The first data set has more variability among its middle 50% of observations. This conclusion concerns central spread, not necessarily the overall range.

How does an outlier-marking box plot differ from a basic box plot?

When outliers are plotted separately as dots, the whiskers generally extend to the smallest and largest non-outlier values rather than to the absolute minimum and maximum.

How are coincident five-number-summary values shown on a box plot?

If two or more summary values are equal, they occupy the same location. For example, if the median equals $Q_3$, there is no separate median line inside the box.

What does it indicate when the minimum equals $Q_1$?

At least 25% of the observations are equal to the minimum value, because the lower quarter has zero numerical width.

Symmetric distribution

A distribution is symmetric when a vertical line can divide its graph into left and right portions that are approximately mirror images.

What relationship is generally expected among the mean, median, and mode in a symmetric unimodal distribution?

The mean, median, and mode are typically close together, often nearly equal, because the data are balanced around the center.

Unimodal versus bimodal distribution

A unimodal distribution has one prominent mode, whereas a bimodal distribution has two modes. In a symmetric bimodal distribution, the two modes can differ from the mean and median.

Left-skewed distribution

A left-skewed, or negatively skewed, distribution has a longer or more influential tail toward smaller values. Low values tend to pull the mean below the median.

Right-skewed distribution

A right-skewed, or positively skewed, distribution has a longer or more influential tail toward larger values. High values tend to pull the mean above the median.

How do the mean and median usually compare in a left-skewed distribution?

The mean is often less than the median because unusually small values pull the mean toward the left tail. The median is less affected by those extreme values.

How do the mean and median usually compare in a right-skewed distribution?

The mean is often greater than the median because unusually large values pull the mean toward the right tail.

Why is the mean usually more affected by skewness and outliers than the median?

The mean uses every numerical value in its calculation, so extreme observations change it substantially. The median depends primarily on the ordered position of the observations.

What is the typical ordering of center measures in the example of a left-skewed distribution?

The mean is often smallest, the median lies between, and the mode is largest: mean $<$ median $<$ mode. This is a tendency, not a universal rule.

What is the typical ordering of center measures in the example of a right-skewed distribution?

The mode is often smallest, the median lies between, and the mean is largest: mode $<$ median $<$ mean.

How can the shape of a distribution be inferred from the relationship between its mean and median?

A mean approximately equal to the median suggests symmetry. A mean below the median suggests left skew, while a mean above the median suggests right skew, although these are general patterns rather than guarantees.

Mean

The mean is the arithmetic average of the data values: $\bar{x}=\frac{\sum x}{n}$ for a sample and $\mu=\frac{\sum x}{N}$ for a population. It is sensitive to extreme values.

Median

The median is the middle value when the data are ordered. If there are two middle values, the median is their average. The median is generally resistant to extreme values.

Mode

The mode is the value or values that occur most frequently in a data set. A data set can have one mode, multiple modes, or no mode.

Standard deviation

Standard deviation is a measure of the typical distance of data values from their mean. It is always nonnegative; a larger value indicates greater overall spread.

What is a deviation from the mean?

For a population, the deviation of $x$ is $x-\mu$; for a sample, it is $x-\bar{x}$. It measures how far, and in which direction, a value lies from the mean.

Why are deviations squared when calculating variance?

Positive and negative deviations would cancel if added directly; their sum is zero. Squaring makes all contributions nonnegative while giving larger deviations greater weight.

Variance

Variance is the average of the squared deviations from the mean. Population variance is denoted $\sigma^2$, and sample variance is denoted $s^2$.

How are standard deviation and variance related?

The standard deviation is the square root of the variance: $\sigma=\sqrt{\sigma^2}$ for a population and $s=\sqrt{s^2}$ for a sample. Variance is expressed in squared units, while standard deviation has the original data units.

What is the formula for the population standard deviation?

For a population of $N$ values, $\sigma=\sqrt{\frac{\sum (x-\mu)^2}{N}}$. With frequencies, use $\sigma=\sqrt{\frac{\sum f(x-\mu)^2}{N}}$.

What is the formula for the sample standard deviation?

For a sample of $n$ values, $s=\sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$. With frequencies, use $s=\sqrt{\frac{\sum f(x-\bar{x})^2}{n-1}}$.

Why does the sample standard deviation divide by $n-1$ instead of $n$?

Using $n-1$ degrees of freedom adjusts for the fact that the sample mean is estimated from the sample. This makes the sample variance a better estimate of population variance.

When should $\sigma$ be used rather than $s$?

Use $\sigma$ when the data represent the entire population and $s$ when the data are a sample from a larger population.

How should frequency data be incorporated into a standard-deviation calculation?

If a value $x$ occurs $f$ times, its squared deviation is weighted by $f$. Thus, use terms such as $f(x-\bar{x})^2$ or $f(x-\mu)^2$ in the numerator.

How does standard deviation help compare two data sets with the same mean?

The data set with the larger standard deviation has values more dispersed around the common mean. The smaller standard deviation indicates greater concentration near the mean.

How can a data value's distance from the mean be expressed in standard deviations?

For a sample, $x=\bar{x}+ks$; for a population, $x=\mu+k\sigma$, where $k$ is the signed number of standard deviations from the mean.

What does a negative standardized distance indicate?

A negative value of $k$ means the observation lies below the mean. For example, $k=-2$ represents a value two standard deviations below the mean.

If the mean wait time is $5$ minutes and the standard deviation is $2$ minutes, how many standard deviations from the mean is a $1$-minute wait?

It is $\frac{1-5}{2}=-2$, so the wait is two standard deviations below the mean.

What does it mean for an observation to be two standard deviations from the mean?

Its absolute distance from the mean is $2$ standard deviations, so $|x-\text{mean}|=2(\text{standard deviation})$. Whether it is unusually far depends on the distribution's shape; two standard deviations is only a rule of thumb.

A sample has mean $10.525$ and sample standard deviation $0.716$. What value is approximately one standard deviation above the mean?

Use $x=\bar{x}+s$: $x\approx10.525+0.716=11.241$, or about $11.24$.

A sample has mean $10.525$ and sample standard deviation $0.716$. What value is approximately two standard deviations below the mean?

Use $x=\bar{x}-2s$: $x\approx10.525-2(0.716)=9.093$, or about $9.09$.