0580 · 9.3Averages and range

Averages and spread basics

0:00A class usually finishes a homework task in a few minutes, but one student takes much longer. That unusually large time can change the mean considerably, while leaving the median unchanged. Both averages are correct, but they describe the results differently. In this video, we will calculate averages and measures of spread, then understand when each is useful. An average describes the centre of a set of data using one representative value.

0:24A measure of spread describes how widely the values are scattered. We need both ideas because groups can have the same average but very different variation. A typical journey time, for example, does not tell us whether that journey is reliable. Our starting skills are addition, division and putting numbers in order. Frequency means how many times a value occurs, or how many observations belong to a group. Keep the units attached to your answers throughout.

0:49An average of times is measured in minutes, and the spread of those times is also measured in minutes.

Mean, median, mode and extremes

0:54Our first exam-style example uses the eleven homework completion times on the board. The question asks us to calculate the mean, median, mode, quartiles, range and interquartile range. The values are already in ascending order, from smallest to largest. Sorting is essential for finding the median and quartiles, because these depend on position. The mean equals the sum of the values divided by the number of values. Imagine combining everyone's completion time into one total, then sharing that total equally among the students.

1:24The amount assigned to each student would be the mean. This explains why every observation contributes to its calculation. Adding all eleven times gives a total of eighty-eight minutes. There are eleven students, so we divide eighty-eight by eleven. The mean is eight minutes. This is an exact answer, and it does not mean that every student actually took eight minutes. The median is the middle value when the data are arranged in order.

1:49With eleven observations, five can sit on either side of the middle one. That makes the sixth observation the median. The sixth value on the board is six, so the median completion time is six minutes. The mode is the value that occurs most frequently. In this example, four occurs twice, while every other value occurs once. The mode is therefore four minutes. Notice that the answer is the data value four, not its frequency of two.

2:15These averages answer different questions about the same data. The mean uses every value and connects directly to the total. The median describes the middle position, so an unusually large value has less influence on it. The mode identifies the most common result, which is useful when choosing the most popular option. An even number of observations leaves two middle values rather than one. For the four values shown, those middle entries are five and eight.

2:40Their sum is thirteen, and dividing by two gives a median of six point five. The median can therefore be a number that does not appear in the original data. A data set can have two modes if two values share the greatest frequency. If every value occurs just once, we normally describe the data as having no mode. The mode can also describe categories, such as a favourite colour.

3:02Arithmetic averages would not make sense for those category names. Suppose the largest time changes from thirty minutes to one hundred and forty minutes. Subtracting thirty from eighty-eight, then adding one hundred and forty, gives a new total of one hundred and ninety-eight. Dividing by eleven gives a mean of eighteen minutes. The median remains six minutes, and the mode remains four minutes. An extreme value is unusually far from the other observations.

3:28Because the mean includes its full numerical size, that value can pull the mean away from most results. The median depends on order rather than the size of the largest observation. For a typical completion time with an extreme result present, the median may therefore be more representative.

Range and interquartile range

3:43Spread gives us information that an average cannot provide. The range is the largest value minus the smallest value. Returning to our original data, 30 − 2 gives twenty-eight minutes. This describes the full distance between the extremes, but uses only two observations. Quartiles help us describe the middle part of the data rather than only the extremes. The lower quartile marks approximately 1/4 of the way through the ordered observations.

4:10The upper quartile marks approximately 3/4 of the way through. We find them here by taking the median of each half. For this odd-sized list, leave the overall median out when forming the two halves. The lower half contains five values, so its middle entry is the third one. That value is four, giving a lower quartile of four minutes. The upper half also contains five entries, with nine in its middle position.

4:34The upper quartile is therefore nine minutes. The interquartile range equals the upper quartile minus the lower quartile. Subtracting four from nine gives five minutes. It measures the spread of approximately the middle fifty per cent of the observations. Counting positions gives the same quartiles, because with eleven values they sit at the third and ninth places. For an even-sized list, split the ordered observations into two equal halves. Find the median of each half, averaging its two middle values when necessary.

5:03Quartiles are data values, not the positions used to locate them. Small data sets do not always divide into four groups containing exactly equal numbers of observations. The range covers the full spread, while the interquartile range describes only the central half. An extreme value can greatly increase the range without changing the interquartile range. The median and interquartile range are therefore often useful together when extreme observations are present.

5:29A smaller spread means the observations are more consistent, not automatically better.

Estimated means from grouped data

5:34Our second exam-style example gives journey times in a grouped frequency table. The first class is the class from zero up to and including ten minutes. The board writes it with inequality signs, as zero is less than t, which is less than or equal to ten. There are twenty-one journeys altogether, but their individual times are no longer available. We cannot calculate the exact mean because the table does not tell us each journey time.

5:58Instead, we represent every observation in a class by its midpoint. The midpoint is halfway between the class endpoints. This gives a reasonable central representative without pretending that we know the original measurements. For the first interval, adding zero and ten gives ten, which we halve to obtain five. The next endpoints add to thirty, so their midpoint is fifteen. For the third class, 50 ÷ 2 gives twenty-five. The final endpoint sum is seventy, giving a midpoint of thirty-five.

6:27Frequency tells us how many observations each midpoint represents. Four journeys in the first class contribute an estimated total of 4 × 5 minutes. We must multiply each midpoint by its frequency before adding. Otherwise, we would give a small group the same influence as a much larger group. The first class contributes twenty minutes to our estimated total. Multiplying seven by fifteen gives one hundred and five for the second group.

6:53The third contribution is 6 × 25, which equals one hundred and fifty. For the last class, 4 × 35 gives one hundred and forty. Adding those four contributions gives an estimated total of four hundred and fifteen minutes. The frequencies add to twenty-one, so divide four hundred and fifteen by twenty-one. The calculator display begins nineteen point seven six one nine. To three significant figures, the estimated mean is nineteen point eight minutes.

7:20The method is estimated total divided by total frequency. In formula form, that means the sum of frequency times midpoint, divided by the sum of frequencies. It remains an estimate even when the arithmetic is performed perfectly. The uncertainty comes from replacing unknown observations with midpoints, not from calculator rounding. The modal class is the group with the greatest frequency. Here, the largest frequency is seven. The modal class is therefore the interval from ten up to and including twenty minutes.

7:51Give the whole interval, because the table cannot identify an exact most frequent journey time.

Grouped discrete data and exam tips

7:57Grouped discrete data use the same midpoint method. Discrete observations are counted in separate values, such as whole numbers of books. This third exam-style example groups ten students by how many books they read. The first group includes zero, one or two books, but we do not know the individual counts. The first class midpoint is one, obtained by halving the endpoint sum of two. For the second class, 3 + 5 = 8, giving a midpoint of four.

8:24The final endpoints add to fourteen, so their midpoint is seven. These are representative counts for the three groups. Multiplying the first midpoint by its frequency gives three. The second group contributes 5 × 4, which equals twenty. For the third group, 2 × 7 gives fourteen. Adding three, twenty and fourteen produces an estimated total of thirty-seven books. The frequencies add to ten students, so 37 ÷ 10 gives an estimated mean of three point seven books.

8:54A mean need not be a whole number, even when every observation is an integer. The class from three to five books inclusive has the largest frequency in the table. That interval is the modal class. Common exam mistakes come from confusing a value with its position or frequency. For a median, count towards the middle only after sorting the data. For quartiles, return the values at the relevant positions, not those position numbers.

9:18For a modal class, report the interval rather than its frequency or midpoint. For grouped means, do not divide by the number of classes. Divide by total frequency because that is the number of observations being represented. Also, do not simply average the midpoints unless every class has the same frequency. Larger groups must contribute more to the estimated total. These methods must be usable without a calculator on Paper two.

9:44On Paper four, a calculator can handle awkward totals and division, but show the midpoint products and total frequency. Unless instructed otherwise, give answers that are not exact to three significant figures. Keep full intermediate values, and retain exact answers such as three point seven without unnecessary rounding. An explanation question may ask why the median is preferable when an extreme time is present. State that the extreme value affects the mean, while the median is less affected by its size.

10:12If asked why a grouped mean is an estimate, explain that the individual values are unknown. Class midpoints have been used to represent them. The mean shares the total equally, the median locates the middle, and the mode identifies the most frequent result. The range measures the full spread, while the interquartile range covers the middle half between the quartiles. Grouped means use frequency-weighted midpoints because individual values are hidden.

10:36The modal class identifies the interval containing the greatest number of observations.