# Measures of relative standing

02/13/2020

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## Outline

• The following topics will be covered in this lecture:
• Z scores
• Percentiles
• Quartiles
• Box plots

## Z scores

• We will start to look at measures of relative standing.

• Measures of relative standing are tools to describe the location of observations in a data set with respect to the other data pieces.
• The most important measure of relative standing is the z score;

• a z score utilizes our understanding of the spread and concentration of normal data in terms of the standard deviation.
• Like the coefficient of variation, we will make this score into a measure on a relative scale so we can compare values from different distributions.

• Specifically, suppose we have a sample value $$x$$ from a normal data set with sample mean $$\overline{x}$$ and sample standard deviation $$s$$.

• Suppose that the population mean is $$\mu$$ and the population standard deviation is $$\sigma$$.

• The z score of $$x$$ is given as

$\begin{matrix} \text{Sample z score} = \frac{x - \overline{x}}{s} & & \text{Population z score} = \frac{x - \mu}{\sigma} \end{matrix}$

• This measures how far $$x$$ deviates from the mean, relative to the size of standard deviation.

• Note, we will typically round the z score to two decimal places.

• Z scores also apply to non-normal data, but their interpretation changes slightly as we cannot use the empirical rule in this context.

## Interpreting z scores Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• Let us recall the empirical rule for normally distributed data:
• Approximately $$68\%$$ of the sample data will lie within one standard deviation $$\sigma$$ of the population mean $$\mu$$, i.e., in $[\mu - \sigma, \mu + \sigma].$
• Approximately $$95\%$$ of sample data will lie within two standard deviations $$2\sigma$$ of the population mean $$\mu$$, i.e., in $[\mu - 2\sigma, \mu + 2\sigma].$
• Approximately $$99.7\%$$ of sample data will lie within three standard deviations $$3\sigma$$ of the population mean $$\mu$$, i.e., in $[\mu - 3\sigma, \mu + 3\sigma].$
• By convention, we will say that an observation is statistically significant low or high in value if there is $$5\%$$ or less chance to observe a value at least as extreme.
• Discuss with a neighbor: if an observation from a normal data set has a z score of $$1$$ is this significant? Why? What is the probability of finding a value at least this extreme?
• By the empirical rule, there is a $$68\%$$ chance of finding a value within one standard deviation.
• Therefore, $$100\% - 68\% = 32\%$$ of values lie outside of one standard deviation – i.e., they are at least this extreme. This is not significant.
• Discuss with a neighbor: if an observation from a normal data set has a z score of $$2$$ is this significant? Why? What is the probability of finding such a value?
• By the empirical rule, there is a $$95\%$$ chance of finding a value within two standard deviations, so $$100\% - 95\% = 5\%$$ of values lie outside of two standard deviations – i.e., they are at least this extreme. This is significant.

### Interpreting z scores continued Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• Less formally, if the data is not normally distributed we will still use the range rule of thumb as an approximation of the empirical rule.
• If the data is not normally distributed, the empirical rule no longer applies, i.e.,
• We are not guaranteed that $$68\%$$ of data will lie within one standard deviation of the mean.
• We are not guaranteed that $$95\%$$ of data will lie within two standard deviations of the mean.
• We are not guaranteed that $$99.7\%$$ of data will lie within three standard deviations of the mean.
• However, Chebyshev’s theorem says that at least $$75\%$$ of data lies within two standard deviations of the mean.
• The actual amount will often be more than this, as this is a lower bound on any data set.
• Therefore, we can still consider a z score of 2 to be interesting for non-normal data, but we must be more careful about our conclusions.

### Interpreting z scores example

• Discuss with a neighbor: which of the following two values is more extreme from the data set from which it came?
1. A baby is born with weight $$4000.0g$$, where the sample data includes $$400$$ babies with sample mean $$\overline{x}=3152.0g$$ and sample standard deviation $$s=693.4g$$
2. An adult is measured with a body temperature of $$99^\circ F$$ out of sample data of $$106$$ adults with sample mean $$\overline{x}=98.20^\circ F$$ and sample standard deviation of $$0.62^\circ F$$.
• To compare these two measurements which exist on different scales and units, we compute their z scores as: $\begin{matrix} \text{baby z score}=\frac{4000.0g - 3152.0g}{693.4g} = 1.22\text{ std} & & \text{heat z score}=\frac{ 99^\circ F - 98.2^\circ F}{0.62^\circ F} = 1.29\text{ std} \end{matrix}$
• By comparing the z scores, we see that the body temperature measurment is more standard deviations away from its sample mean than the baby’s weight.
• Even though the difference in temperature units is small, the relatively small standard deviation in the measurements makes this a more extreme value with respect to its sample data set.
• This illustrates the purpose of the z score, in that it makes all measurements comparable on a relative, standardized scale.
• We note that the z score is signed with a $$\pm$$. One important property of the z score is that it tells whether the value lies above or below the mean.
• In the above both measurements lie above the mean value of the samples and for this reason they are positive;
• on the other hand, whenever we see a negative z score, we know already that the measurement was below the mean of the samples.

## Percentiles

• Percentiles – these are measures of location, denoted $$P_1, P_2,\cdots, P_{99}$$, which divide a set of data into $$100$$ groups with about $$1\%$$ of the values in each group.

• An example we know already is the median.
• Indeed, the median is the $$P_{50}$$ percentile, which separates the data into groups with $$50\%$$ of the data above and $$50\%$$ of the data below.
• There are different ways in which the percentile can be computed, and therefore we will consider one of several possible approaches;

• the important part is to understand how we can convert a data value into a percentile, and
• how to convert a percentile back into a data value.
• We will discuss both of these in the following, but note,

• converting back and forth, the results can be inconsistent.
• We should be careful therefore about what is the question at hand.

### Converting data into percentiles

• Suppose we have samples given as $$x_1, \cdots, x_n$$ where $$n$$ is the total number of samples in the data set.

• Suppose the measurements are quantitative, so that we can arrange the samples in order;

• that is, up to re-naming samples, we can write $x_i \leq x_{i+1}$ for each $$i = 1,\cdots, n-1$$.
• Then, for a particular value $$x$$, its percentile can be computed as,

\begin{align}\text{Percentile of }x &= \frac{\text{Number of samples with value less than } x}{\text{Number of total samples}}\times 100 \end{align}

• If we can order the sample values as above, we thus look for the index $$i$$ for which $x_i < x \leq x_{i+1}$
• That is, we count the number of samples $$i$$ with value strictly less than $$x$$;
• the next ordered sample $$x_{i+1}$$ can have a value that is either greater than or equal to the value $$x$$.
• If we choose the index $$i$$ as above, the formula becomes \begin{align}\text{Percentile of }x &= \frac{i}{n}\times 100 \end{align}

### Finding the percentile of some value Courtesy of Mario Triola, Essentials of Statistics, 5th edition

• In the table to the left, we see an example data set where the samples have been ordered from low to high in the value.
• There are $$4$$ rows and $$10$$ columns to this table.
• The samples are the number of chocolate chips in a batch of 40 cookies.
• Discuss with a neighbor: what is the percentile of $$x=23$$? That is, what is the percent of samples that have value lower than $$23$$, relative to the total number of samples?
• Notice there are $$10$$ columns and the first row consists of samples with value less than $$23$$.
• That is to say, $x_{10} < 23 \leq x_{11}.$
• In this regard, we have, $\text{Percentile of }23 = \frac{10}{40}\times 100 = 25.$
• Therefore, we say $$x=23$$ is in the $$25$$-th percentile.
• Similar to the median, we can say that a cookie with $$23$$ chips approximately separates the cookies with the lowest $$25\%$$ of chips from those with the highest $$75\%$$ of chips.
• Note: we do not say $$P_{25}=23$$, we will show how to find $$P_{25}$$ in the following.