FS4, HS5 - Averages and range

FS4, HS5 - Averages and range

Calculate mean, median, mode and range from raw data, use summary conditions to recover missing information, and compare distributions in context. This is the shared central-tendency and range content of FS4 and HS5.

Mean, median and mode

Can two groups have the same “average” but behave very differently? First decide which average is meant and what information it keeps.

Three measures of centre

The mean shares the total equally: add all values and divide by their number. The median is the middle value after ordering from smallest to largest; with an even number, average the two middle values. The mode is the value occurring most often.

For nn ordered values, the median is in position (n+1)/2(n+1)/2 when nn is odd. For even nn, use positions n/2n/2 and n/2+1n/2+1. These are positions in the ordered list, not values to report as the median.

Worked example 1: an odd number of observations

Question. Five pupils read 10,2,1,5,210,2,1,5,2 books. Find the mean, median and mode.

Plan. Use the original values for the total, and an ordered list for the middle and repeated values.

mean=10+2+1+5+25=205=4.\text{mean}=\frac{10+2+1+5+2}{5}=\frac{20}{5}=4.

In order: 1,2,2,5,101,2,2,5,10. The third value is 22. The value 22 occurs twice; every other value occurs once.

Answer. Mean 44 books, median 22 books, mode 22 books.

Check. 4×5=204\times5=20, recovering the total. Two observations lie on either side of the middle position. The mean need not be an observed value.

Keep the same procedures when values are negative, and identify both middle positions for an even-sized list.

Worked example 2: an even list with negative values

Question. Six temperatures, in ∘C^{\circ}\mathrm C, are −1,−5,2,−3,0,−5-1,-5,2,-3,0,-5. Find the three averages.

Plan. Order the signed values correctly, then use positions 33 and 44 for the median.

The ordered list is −5,−5,−3,−1,0,2-5,-5,-3,-1,0,2.

mean=−1−5+2−3+0−56=−126=−2median=−3+(−1)2=−2.\begin{aligned} \text{mean}&=\frac{-1-5+2-3+0-5}{6}=\frac{-12}{6}=-2\\ \text{median}&=\frac{-3+(-1)}{2}=-2. \end{aligned}

Answer. Mean −2 ∘C-2\,^\circ\mathrm C, median −2 ∘C-2\,^\circ\mathrm C, mode −5 ∘C-5\,^\circ\mathrm C.

Check. −2×6=−12-2\times6=-12. On a number line, −2-2 is halfway between the middle values −3-3 and −1-1.

If all values occur once, report no mode. If several values tie for the greatest repeated frequency, report all those modes: 1,1,3,3,71,1,3,3,7 has modes 11 and 33. A mode can also describe categorical data, such as the most popular fruit; a numerical mean cannot average fruit names.

To reverse a mean calculation, recover the total:

Mean and total

mean=total of valuesnumber of values,total=mean×number of values.\text{mean}=\frac{\text{total of values}}{\text{number of values}}, \qquad \text{total}=\text{mean}\times\text{number of values}.

These are methods to know; they are not supplied formulas on the formula sheet.

Use the appropriate count before and after a new observation.

Worked example 3: an added value from two means

Question. Three scores have mean 88. A fourth score is added and the new mean is 1010. Find the fourth score.

Plan. Work with the two totals, since subtracting the means would ignore the different counts.

old total=3×8=24,new total=4×10=40.\text{old total}=3\times8=24,\qquad \text{new total}=4\times10=40.

The added score is 40−24=1640-24=16.

Answer. 1616.

Check. One possible original set is 6,8,106,8,10. Adding 1616 gives total 4040, so the new mean is 40/4=1040/4=10.

Summary conditions can also determine an entire short data set. Use order to make their relationships visible. The range is largest value minus smallest value; we will examine this measure of spread more closely in the next part.

Worked example 4: constructing data from conditions

Question. Four whole numbers have mode 66, median 55 and range 55. Give a possible set.

Plan. Repeat the mode and arrange the four numbers as a,b,6,6a,b,6,6. Use the middle pair and the extreme pair.

For the median, (b+6)/2=5(b+6)/2=5, so b+6=10b+6=10 and b=4b=4. For the range, 6−a=56-a=5, so a=1a=1.

Answer. 1,4,6,61,4,6,6.

Check. 66 alone occurs twice; the middle pair averages to (4+6)/2=5(4+6)/2=5; largest minus smallest is 6−1=56-1=5. All four values are whole numbers.

Check every condition separately: satisfying the median alone does not guarantee the required mode or range.

Watch out

For the median, order first and average the middle values when there are two. For the mean, divide by the number of observations, including repeated ones.

Calculate the three averages for a fresh even-sized list.

Range and outliers

An average describes a centre. It does not tell us how widely the observations vary.

Range and outlier

The range is largest value minus smallest value. It measures spread in the same units as the data. An outlier is a value unusually far from the rest; whether it is surprising depends on the context.

For example, values from −8 ∘C-8\,^\circ\mathrm C to 3 ∘C3\,^\circ\mathrm C have range 3−(−8)=11 ∘C3-(-8)=11\,^\circ\mathrm C. Do not subtract the extremes in the opposite order.

The mean uses every value's size, so an extreme value can pull it substantially. The median depends mainly on order and middle positions, so it is usually less affected. The range uses only the two extremes, so it can change greatly.

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