Math Core

A graph shows the shape of a distribution; numbers pin down its center and spread precisely. This lesson covers the summary statistics you'll use all year in AP Statistics: the mean and median for center; the standard deviation, range and interquartile range for spread; and percentiles for describing where one value sits within a distribution.

Statistics and parameters

A number that describes a sample is called a statistic. A number that describes an entire population is called a parameter. For now you'll mostly compute statistics from data in hand. The notation reflects the difference: the sample mean is $\bar{x}$ and the sample standard deviation is $s_x$ , while the population mean and standard deviation are written $\mu$ and $\sigma$ .

Measures of center

The mean of $n$ values $x_1, x_2, \dots, x_n$ is

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum x_i}{n}.

The symbol $\sum$ (capital sigma) means "add up." The median is the middle value of the ordered data: the value in position $\dfrac{n + 1}{2}$ . When $n$ is even, it is the average of the two middle values.

The mean is the balance point of the distribution. Because it uses the size of every value, a single extreme value can pull it a long way. The median depends only on the order of the values, so extreme values barely affect it. We say the median is resistant and the mean is not resistant.

Mean versus median and shape

Roughly symmetric: mean $\approx$ median.
Skewed right: the long right tail pulls the mean above the median.
Skewed left: the long left tail pulls the mean below the median.

For skewed data or data with outliers, the median is usually the better measure of a typical value.

Measures of spread

The range is $\text{maximum} - \text{minimum}$ . It is easy to compute but depends only on the two most extreme values.

The quartiles split the ordered data into quarters. $Q_1$ is the median of the values to the left of the overall median, and $Q_3$ is the median of the values to its right. (When $n$ is odd, leave the median itself out of both halves.) The interquartile range is

\text{IQR} = Q_3 - Q_1,

the spread of the middle half of the data. The IQR is resistant.

The standard deviation measures the typical distance of the values from the mean.

Definition

Sample standard deviation

s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

Each $x_i - \bar{x}$ is a deviation from the mean. The quantity under the square root, $s_x^2$ , is the variance. The standard deviation has the same units as the data. It is $0$ only when every value is identical, and it is not resistant: one outlier can inflate it a lot.

Why $n - 1$ instead of $n$ ? The deviations always add to $0$ , so once you know $n - 1$ of them, the last one is determined. There are only $n - 1$ "free" deviations, called the degrees of freedom. Dividing by $n - 1$ also makes $s_x^2$ a better estimate of the population variance. Your calculator reports both $s_x$ and $\sigma_x$ (which divides by $n$ ); in AP Statistics, use $s_x$ for sample data.

Worked example: Mean, median and standard deviation

A barista timed how many minutes six drink orders took: $8, 11, 12, 14, 15, 18$ . Find the mean, median and standard deviation, and interpret the standard deviation.

Mean. $\bar{x} = \dfrac{8 + 11 + 12 + 14 + 15 + 18}{6} = \dfrac{78}{6} = 13$ minutes.

Median. The two middle values are $12$ and $14$ , so the median is $13$ minutes.

Standard deviation. The deviations are $-5, -2, -1, 1, 2, 5$ (they add to $0$ ). Their squares are $25, 4, 1, 1, 4, 25$ , which add to $60$ . So

s_x = \sqrt{\frac{60}{6 - 1}} = \sqrt{12} \approx 3.46 \text{ minutes}.

Interpretation. The time to make a drink order typically varies by about $3.46$ minutes from the mean of $13$ minutes.

That last sentence is the standard AP interpretation of a standard deviation: "The [variable] typically varies by about [s] [units] from the mean of [x̄]."

Worked example: Quartiles and IQR

Find the quartiles and IQR of these $11$ quiz scores:

52, \ 61, \ 64, \ 68, \ 70, \ 73, \ 75, \ 78, \ 81, \ 84, \ 96

The median is the 6th value, $73$ . The lower half is $52, 61, 64, 68, 70$ , so $Q_1 = 64$ . The upper half is $75, 78, 81, 84, 96$ , so $Q_3 = 81$ . The IQR is $81 - 64 = 17$ points: the middle half of the scores span $17$ points.

Percentiles and cumulative relative frequency

The $p$ th percentile of a distribution is the value with $p\%$ of the data less than or equal to it. If your score is at the $80$ th percentile, about $80\%$ of the scores are at or below yours. The median is the $50$ th percentile, $Q_1$ is about the $25$ th, and $Q_3$ is about the $75$ th.

A cumulative relative frequency graph plots, for each value $x$ , the proportion of the data less than or equal to $x$ . It lets you read percentiles in both directions: from a value up to its percentile, or from a percentile across to its value.

Worked example: Reading a cumulative relative frequency graph

The graph shows the cumulative relative frequency of the shopping times of $40$ store customers. (For example, $22.5\%$ of the customers shopped for $20$ minutes or less.)

Cumulative relative frequency of shopping time (minutes)Open in grapher →

At what percentile is a shopping time of $40$ minutes?
Estimate the median shopping time.

Solutions.

Go up from $40$ to the graph: the height is $0.75$ . A $40$ -minute trip is at about the 75th percentile.
Go across from $0.5$ to the graph. It crosses between $20$ minutes (height $0.225$ ) and $30$ minutes (height $0.525$ ), close to $30$ . Assuming the times are spread evenly within that interval, the median is about $20 + 10 \cdot \dfrac{0.5 - 0.225}{0.525 - 0.225} \approx 29$ minutes.

Changing units: linear transformations

Converting units applies the same rule to every value, such as $F = 1.8C + 32$ for temperatures.

Adding a constant $a$ to every value adds $a$ to measures of center and location (mean, median, quartiles, percentiles) but does not change measures of spread (range, IQR, standard deviation).
Multiplying every value by a positive constant $b$ multiplies measures of center, location and spread by $b$ .
Neither changes the shape of the distribution.

Worked example: Celsius to Fahrenheit

The daily high temperatures in a city one spring had a mean of $20^\circ\text{C}$ and a standard deviation of $4^\circ\text{C}$ . Find the mean and standard deviation in degrees Fahrenheit.

Mean: $1.8 \cdot 20 + 32 = 68^\circ\text{F}$ . Standard deviation: multiplying by $1.8$ scales the spread, and adding $32$ doesn't change it, so $1.8 \cdot 4 = 7.2^\circ\text{F}$ .

Common mistake

Two common slips: dividing by $n$ instead of $n - 1$ when computing $s_x$ by hand, and adding the constant to the standard deviation during a transformation. Shifting all values moves the whole distribution, but the distances between values, and so the spread, stay the same.

Practice

Practice 1

Find the sample standard deviation $s_x$ of $4, \ 8, \ 9, \ 11, \ 13$ . Round to the nearest hundredth.