A Complete Summary of Six Quiz Scores

One Small Data Set, Several Useful Summaries

A single set of scores can be described in several complementary ways. A measure of center gives a typical value; measures of spread describe how far observations vary; standardized scores express positions relative to the data set; and percentile ranks describe how many scores lie below a given score. No single number captures all of these features.

We will summarize the six quiz scores \(4, 5, 5, 7, 9,\) and \(12\). These are hypothetical scores, measured in points. The observations are already arranged from smallest to largest, and the two scores of 5 are a tie. The calculations below describe these six scores; they do not, by themselves, support a conclusion about students or quizzes beyond this data set.

Convention: For the quartiles in this example, divide the ordered data into a lower half and an upper half, then take the median of each half. For percentile ranks with ties, use the midrank convention: the rank percentage for a value is the percentage of observations below it plus half the percentage tied at it. Other percentile conventions can give different results, so the convention should be stated.

Center: Mean and Median

The sample mean is the sum of the observed values divided by their number. It uses every score, so an unusually high or low score can affect it. The median is the middle value after ordering the observations. With an even number of observations, it is the average of the two middle values.

Worked Example: Finding the Mean and Median

For the scores \(4, 5, 5, 7, 9, 12\), there are \(n=6\) observations. Their sum is \(42\), so the mean is

\bar{x}=\frac{4+5+5+7+9+12}{6} =\frac{42}{6}=7\text{ points}.

To find the median, locate the two middle positions. Because \(n=6\), these are positions 3 and 4, containing 5 and 7. Thus

\text{Median}=\frac{5+7}{2}=6\text{ points}.

The mean is 7 points and the median is 6 points. They are not expected to be identical: the mean uses all six scores, while the median is determined by the central positions. Here, the higher score of 12 pulls the mean upward relative to the median.

The location of the median follows directly from the ordering. The first three values, \(4,5,5\), lie below the pair of central positions, and \(9,12\) lie above them; averaging the central pair gives a value that divides the ordered data into two equal-sized groups. This is why the even-sample median is not simply the third or fourth score.

Quartiles and Interquartile Range

Quartiles divide ordered data into sections. Under the median-of-halves convention used here, \(Q_1\) is the median of the lower half and \(Q_3\) is the median of the upper half. The interquartile range, or IQR, is their difference, \(Q_3-Q_1\). It measures the spread of the middle half of the observations and has the same units as the data.

Worked Example: Finding the Quartiles and IQR

With six ordered scores, the lower half is \(4,5,5\), and the upper half is \(7,9,12\). The middle value of the lower half is 5, and the middle value of the upper half is 9. Therefore,

Q_1=5\text{ points},\qquad Q_3=9\text{ points}.

Subtract the first quartile from the third:

\mathrm{IQR}=Q_3-Q_1=9-5=4\text{ points}.

Using this convention, the first quartile is 5 points, the third quartile is 9 points, and the interval from 5 to 9 points describes the central portion of the ordered scores. A different software percentile convention may produce different quartile values, especially in small data sets. The method is therefore part of a complete report.

Sample Standard Deviation

The sample standard deviation \(s\) summarizes the typical distance of observations from the sample mean. First find each deviation \(x_i-\bar{x}\), square the deviations, add them, divide by \(n-1\) to obtain the sample variance, and take the square root. Since deviations are squared before being added, the variance is in squared points; taking its square root returns the standard deviation to points.

Sample standard deviation: For observations \(x_1,\ldots,x_n\), the sample standard deviation is

s=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}.

It is a descriptive measure of spread around the sample mean. The division by \(n-1\) is the sample-variance convention.

Worked Example: Calculating the Sample Standard Deviation

The mean is 7 points. The deviations from the mean are \(-3,-2,-2,0,2,5\) points, and their squares are \(9,4,4,0,4,25\) squared points. Their sum is \(46\) squared points. Substituting into the sample standard deviation formula gives

s=\sqrt{\frac{46}{6-1}} =\sqrt{9.2} \approx 3.033\text{ points}.

The sample variance is \(9.2\) squared points, while the sample standard deviation is approximately \(3.033\) points. The standard deviation is not the range and is not the average of the absolute distances; it is calculated from squared deviations about the mean.

Z Scores: Positions in Standard-Deviation Units

A z score expresses how far a value is from a mean in standard-deviation units. For this descriptive summary, we standardize each score using the sample mean and sample standard deviation:

z=\frac{x-\bar{x}}{s}.

A positive z score means the score is above the mean; a negative z score means it is below. A z score of 0 means it equals the mean. Z scores have no units because the numerator and denominator are both measured in points.

Worked Example: Standardizing All Six Scores

For instance, the score of 12 points has z score

z=\frac{12-7}{3.033}\approx 1.648.

It is about 1.648 sample standard deviations above the mean. Applying the same calculation to every distinct score gives:

Score \(x\) (points)	Calculation \((x-7)/3.033\)	Z score
4	\((4-7)/3.033\)	\(-0.9891\)
5	\((5-7)/3.033\)	\(-0.6594\)
5	\((5-7)/3.033\)	\(-0.6594\)
7	\((7-7)/3.033\)	0
9	\((9-7)/3.033\)	0.6594
12	\((12-7)/3.033\)	1.648

The two tied scores have the same z score because they have the same distance from the mean. These are descriptive z scores calculated from this sample’s own mean and standard deviation. They do not imply that the scores follow a normal distribution.

Percentile Ranks and Ties

A percentile rank describes the relative standing of an observed value: it reports the percentage of observations below that value, with a stated rule for ties. Here we use the midrank convention. If \(L\) observations are lower than a value and \(E\) observations are equal to it, its percentile rank is

100\left(\frac{L+\tfrac{1}{2}E}{n}\right)\%.

Half of the tied observations are counted toward the rank. This gives tied values the same percentile rank and avoids arbitrarily placing one tied observation ahead of another. A percentile rank is not the same as a percentile value: the rank is a percentage associated with a value, while a percentile value is a value associated with a specified percentage position.

Worked Example: Percentile Ranks for Selected Scores

For a score of 5, one observation is below it (the score of 4), and two observations equal it. Thus \(L=1\), \(E=2\), and \(n=6\):

100\left(\frac{1+\tfrac{1}{2}(2)}{6}\right) =100\left(\frac{2}{6}\right) \approx 33.33\%.

So a score of 5 has a midrank percentile rank of about 33.33%. For a score of 9, four observations are below it and one equals it:

100\left(\frac{4+\tfrac{1}{2}(1)}{6}\right) =100\left(\frac{4.5}{6}\right)=75.00\%.

For the maximum score of 12, five observations are below it and one equals it, giving \(100(5.5/6)\approx91.67\%\). The maximum does not receive a 100% midrank under this convention because half of its own tied group is included, rather than all of it. These percentages describe rank within these six scores, not the probability of a score in a larger population.

Putting the Summary Together

A compact report can collect the measures while keeping their definitions and units clear. The following table summarizes the six hypothetical quiz scores under the stated quartile and tie conventions.

Feature	Summary	Interpretation for these scores
Mean	7 points	Arithmetic average of all six scores
Median	6 points	Average of the two central ordered scores
First quartile \(Q_1\)	5 points	Median of the lower three scores
Third quartile \(Q_3\)	9 points	Median of the upper three scores
Interquartile range	4 points	Difference \(Q_3-Q_1\)
Sample standard deviation \(s\)	3.033 points	Spread around the sample mean, in points
Z score for 12 points	1.648	About 1.648 sample standard deviations above the mean
Percentile rank for 5 points	33.33%	Midrank among these six scores

The measures answer different questions. The mean and median summarize center, but the mean is more affected by extreme values. The IQR focuses on the middle portion of the ordered data; the standard deviation uses every observation’s distance from the mean. Z scores and percentile ranks describe relative standing, but use different scales: z scores measure standardized distance, whereas percentile ranks express a percentage position.

Common Mistakes

Using the wrong middle values. With six observations, the median is the average of positions 3 and 4, not just the third observation.
Mixing quartile conventions. Quartiles depend on the stated calculation rule in a small data set. State the median-of-halves rule rather than presenting its results as the only possible software output.
Dividing by \(n\) for sample standard deviation. The sample standard deviation uses \(n-1\) in the denominator of the variance formula; for these scores that is 5, not 6.
Dropping the tie adjustment. Under the midrank rule, both scores of 5 have the same percentile rank, with half of the tied group included.
Confusing a percentile rank with a percentile value. “5 points has a rank of 33.33%” is a rank statement under the chosen rule; it does not say that the 33.33rd percentile value is 5 points under every percentile convention.
Giving a z score units. The score and standard deviation are both in points, so their ratio is unit-free.

Key takeaway: A complete numerical summary is meaningful only when calculations, units, and conventions are clear. For these six scores, the mean is 7 points, the median is 6 points, the median-of-halves quartiles are 5 and 9 points, the IQR is 4 points, and the sample standard deviation is about 3.033 points.

Check Your Understanding

Use the scores \(4,5,5,7,9,12\), and show your calculations where appropriate.

Why is the median the average of 5 and 7 rather than just one of those scores?
Using the median-of-halves convention, what are the lower half, upper half, and IQR?
What denominator is used to calculate the sample variance here, and why is the standard deviation measured in points rather than squared points?
Calculate the z score for the score of 4 using \(\bar{x}=7\) and \(s\approx3.033\). Interpret its sign and approximate size.
Under the midrank convention, calculate the percentile rank of 7 points. How many scores are below 7, and how many are tied at 7?