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

Significance of measurements by z score.

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

Significance of measurements by z score.

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

Table of sample values ordered low to high.

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.

Finding the value corresponding to some percentile