02/13/2020

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

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**.

- Measures of relative standing are tools to
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**.

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.

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.

- We

- 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**.

**Discuss with a neighbor:**which of the following two values is more extreme from the data set from which it came?- 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 \( s=693.4g \)
- 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**– 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.

- An example we know already is the
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**.

- the important part is to understand
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.

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 \[ \begin{align}\text{Percentile of }x &= \frac{i}{n}\times 100 \end{align} \]

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.