## 13.2 Computing the average value

The average (or *location*, or *centre*, or *typical value*) for
*quantitative sample data* can be described
in many ways;
the two most common ways are:

- the
**sample mean**(or*sample arithmetic mean*), which estimate the population mean; and - the
**sample median**, which estimates the population median.

In both cases,
the population parameter is
*estimated* by a sample statistic.
Understanding
whether to use the mean or median is important.

**Think 13.1 (Difference between averages) **Consider the daily river flow volume (called ‘streamflow’) at the Mary River from 01 October 1959 to 17 January 2019,
summarised by month in Table 13.1
(from Queensland DNRM).

The ‘average’ daily streamflow in February could be quoted using either the mean or the median;
but the two give *very* different values for the ‘average’:

- the
*mean*daily flow is 1123.2ML. - the
*median*daily flow is 146.1ML.

Month | Mean | Median |
---|---|---|

Jan | 849.3 | 71.3 |

Feb | 1123.2 | 146.1 |

Mar | 793.9 | 194.9 |

Apr | 622.5 | 141.7 |

May | 348.4 | 118.4 |

Jun | 378.7 | 83.6 |

Jul | 259.3 | 68.8 |

Aug | 108.6 | 55.5 |

Sep | 100.9 | 48.0 |

Oct | 151.2 | 37.6 |

Nov | 186.6 | 45.3 |

Dec | 330.8 | 64.1 |

### 13.2.1 Computing the average: The mean

The mean of the population is denoted by \(\mu\), and its value is almost always unknown.

Instead, the mean of the population is *estimated* by the mean of the sample,
which is denoted by \(\bar{x}\) (an \(x\) with a line above it).
In this context,
the unknown *parameter* is \(\mu\),
and the *statistic* is \(\bar{x}\).
The sample mean is used to *estimate* the population mean.

The Greek letter \(\mu\) is pronounced ‘myoo,’ as in **mu**sic.

**Example 13.2 (A small data set to work with) **To demonstrate ideas,
consider a small data set
for answering this descriptive RQ:

For mature Jersey cows, what is the average percentage butterfat in their milk?

The *population* is ‘milk from Jersey cows,’
and an estimate of the population mean percentage butterfat is sought.
The population mean is denoted by \(\mu\).

*sample*is studied (Sokal and Rohlf 1995; Hand et al. 1996): The unknown population mean is estimated using the sample mean (\(\bar{x}\)). Measurements were taken from milk from 10 cows, in percentages (Table 13.2).

4.8 | 5.2 | 5.2 | 5.4 | 5.2 |

6.5 | 4.5 | 5.7 | 4.8 | 5.2 |

The *sample mean* is what people usually think of as the ‘average.’
The sample mean is actually the ‘balance point’ of the
observations. The animation below shows how the mean acts as the balance point.
Alternatively,
the mean is the value
such that the positive and negative distances of the observations from the mean add to zero
, as shown in the animation below.
Both of these explanations seem reasonable
for identifying the ‘average’ of the data.

**Definition 13.3 (Mean)**The

**mean**is one way to measure the ‘average’ value of quantitative data. The

*arithmetic mean*can be considered as the ‘balance point’ of the data, or the value such that the positive and negative distances from the mean add to zero.

To find the *value* of the sample mean:

*Add*(shown using the symbol \(\sum\)) all the observations (denoted by \(x\)); then*Divide*by the number of observations (denoted by \(n\)).

In symbols: \[ \bar{x} = \frac{\sum x}{n}. \] This means to add up (indicated by \(\sum\)) the observations (denoted by \(x\)), then divide by the size of the sample (denoted by \(n\)).

**Example 13.3 (Computing a sample mean)**For data for the Jersey cow data (Example 13.2), an estimate of the population mean percentage butterfat is found using the sample information: sum all \(n=10\) observations and divide by \(n\): \[\begin{align*} \overline{x} &= \frac{\sum x}{n} = \frac{4.8 + 6.5 + \cdots + 5.2}{10}\\ &= \frac{52.5}{10} = 5.25. \end{align*}\] The sample mean, the best estimate of the population mean, is 5.25

*percent*.

*Statistics Mode*) will be used to compute the sample mean. However, knowing

*how*these quantities are computed is important.

**Think 13.2 (Mean)**For the butterfat data (Table 13.2), what is the value of \(\mu\), the

*population*mean?

*sample*mean, but not the

*population*mean. We only have an

*estimate*of the value of the population mean.

**Think 13.3 (Estimating a mean)**A study of eyes (Ehlers 1970) aimed to estimate the average thickness of eyes affected by glaucoma. The collected data (in microns) are shown in Table 13.3. Estimate the population mean corneal thickness.

484 | 492 | 436 | 464 |

478 | 444 | 398 | 476 |

Software and calculators often produce numerical answers to many decimal places, some of which may not be meaningful or useful. A useful rule-of-thumb is to round to one or two more significant figures than the original data.

For example, the butterfat data are given to one decimal place. The*sample mean*weight can be given to two decimal places: \(\bar{x}=5.25\)%.

### 13.2.2 Computing the average: The median

The median is a value separating the larger half of the data from the smaller half of the data.
In a data set with \(n\) values,
the median is *ordered* observation number \(\displaystyle \frac{n+1}{2}\).
The median is:

**not**equal to \(\displaystyle \frac{n+1}{2}\).**not**halfway between the minimum and maximum values in the data.

Most calculators cannot find the median.

**Definition 13.4 (Median)**The

**median**is one way to measure the ‘average’ value of some data. The

*median*is a value such that half the values are larger than the median, and half the values are smaller than the median.

**Example 13.4 (Find a sample median)**To find the sample median for the Jersey cow data (Example 13.2), first arrange the data

*in numerical order*(Table 13.4). The median separates the larger 5 numbers from the smaller 5 numbers. With \(n=10\) observations, the median is the ordered observation located between the fifth and sixth observations (i.e., at position \((10+1)/2 = 5.5\); the

*median itself is not 5.5*). So the sample median is between \(5.2\) (ordered observation five) and \(5.2\) (ordered observation six): the sample median is \(5.20\) percent.

4.5 | 4.8 | 4.8 | 5.2 | 5.2 |

5.2 | 5.2 | 5.4 | 5.7 | 6.5 |

*sample*median, but not the

*population*median. We only have an

*estimate*of the value of the population median.

**Think 13.5 (Medians) **A study of eyes
(Ehlers 1970)
aimed to estimate
the average thickness of eyes affected by glaucoma.

*median*corneal thickness. What is the population median?

With \(n=8\) observations,
the median is ordered observation number \((8+1)/2 = 4.5\),
halfway between ordered observation numbers 4 and 5.
After sorting into increasing order,
the two middle numbers (the 4th and 5th) are
464 and 476.
The median could be *any* number between 464 and 476,
but the usual answer would be that the median is \((464 + 476)/2 = 470\).

*sample*median is 470 microns; the value of the

*population*median remains unknown.

To clarify:

- If the sample size \(n\) is
*odd*, the median is the middle number when the observations are ordered. - If the sample size \(n\) is
*even*(such as in Think 13.5), the median is halfway between the two middle numbers, when the observations are ordered.

Some software uses different rules when \(n\) is even.

### 13.2.3 Which average to use?

Consider again estimating the average daily streamflow at the Mary River (Bellbird Creek)
during February
(Table 13.1):
The *mean* daily streamflow is 1123.2ML,
and the *median* daily streamflow is 146.1ML.
Which is the ‘best’ average to use?

A dot chart of the daily stream flow
(Fig. 13.2)
shows that the data are *very* highly right-skewed,
with many *very* large outliers:
the maximum value is
156586.4ML,
more than one hundred times larger than the mean of 1123.2ML).
In fact,
about
86% of the observations are *less* than the mean.
In contrast,
about 50% the values are less than the median (by definition).
For these data,
the mean is hardly a *central* value…

The streamflow data are very highly skewed (to the right), which is important and relevant:

*Means*are best used for approximately symmetric data: the mean is influenced by outliers and skewness.*Medians*are best used for data that are skewed or contain outliers: the median is not influenced by outliers and skewness.

Means tend to be too large if the data contains large outliers or severe right skewness, and too small if the data contains small outliers or severe left skewness.

For the Mary River data,
the large outliers—and the fact that they are so *extreme* and abundant—result in the
mean being substantially influenced by the outliers,
which explains why the mean is much larger than the median.
*The median is the better measure of average for these data.*

The mean is generally used if possible
(for practical and mathematical reasons),
and is the most commonly-used measure of location.
However,
the mean *is* influenced by outliers and skewness;
the median *is not* influenced by outliers and skewness.
The mean and median are similar in approximately symmetric distributions.
Sometimes, quoting *both* the mean and the median may be appropriate.

**Think 13.6 (Which average to use) **An engineering study (Hald 1952)
was studying a new building material
to determine the average permeability time.