4.4 Standard Normal Distribution and z-Scores
===============================================

The purpose of this section is to build a **standard reference scale** for
continuous probability calculations.

We will focus on the standard normal random variable :math:`Z` and on using
:math:`z`-scores. The general normal distribution will be introduced in the
**next section**.

Key Terms
---------

- **standard normal random variable**
- **z-score**
- **standardization**
- **cumulative probability**
- **left-tail probability**
- **right-tail probability**
- **standard normal table**
- **symmetry**

4.4.1 Learning Objectives
-------------------------

After completing this section, you should be able to:

1. explain what a :math:`z`-score represents;
2. compute a :math:`z`-score from a value, mean, and standard deviation;
3. interpret positive, negative, and zero :math:`z`-scores;
4. recognize the standard normal reference variable :math:`Z`;
5. use a cumulative standard normal table;
6. calculate left-tail, right-tail, and interval probabilities for :math:`Z`;
7. use symmetry and complements to simplify calculations;
8. find approximate :math:`z`-values from cumulative probabilities.

4.4.2 The z-Score: A Standard Distance
--------------------------------------

Suppose a quantity has mean :math:`\mu` and standard deviation :math:`\sigma`.
For an observed value :math:`x`, define its **z-score** as

.. math::

   z=\frac{x-\mu}{\sigma}.

A :math:`z`-score measures the distance from the mean in **standard deviation
units**.

- :math:`z>0`: the value is above the mean;
- :math:`z<0`: the value is below the mean;
- :math:`z=0`: the value equals the mean;
- :math:`|z|` tells how many standard deviations the value is from the mean.

For example, suppose a measurement has

.. math::

   \mu=10,
   \qquad
   \sigma=2.

If :math:`x=13`, then

.. math::

   z
   =
   \frac{13-10}{2}
   =
   1.5.

Thus, 13 is **1.5 standard deviations above the mean**.

If :math:`x=8`, then

.. math::

   z
   =
   \frac{8-10}{2}
   =
   -1.

Thus, 8 is **1 standard deviation below the mean**.

.. note::

   A :math:`z`-score has no physical unit. The original unit cancels when
   dividing by the standard deviation.

The relationship can also be reversed. From

.. math::

   z=\frac{x-\mu}{\sigma},

we obtain

.. math::

   x=\mu+z\sigma.

This form will become useful when we later convert a standardized value back to
an original measurement scale.

4.4.3 Standardization Does Not Automatically Mean Normality
-------------------------------------------------------------

The calculation

.. math::

   z=\frac{x-\mu}{\sigma}

can be used to describe the relative position of a value on many different
measurement scales.

However, computing a :math:`z`-score does **not** by itself imply that the
underlying variable has a normal distribution.

In this section, we separately study a specific reference random variable,
called the **standard normal random variable**.

The next section will explain when an original random variable can be connected
to this standard normal reference curve for probability calculations.

4.4.4 The Standard Normal Random Variable
------------------------------------------

A **standard normal random variable** is denoted by

.. math::

   Z.

It has

.. math::

   E(Z)=0

and

.. math::

   \operatorname{SD}(Z)=1.

Equivalently,

.. math::

   \operatorname{Var}(Z)=1.

The usual notation is

.. math::

   Z\sim N(0,1).

For now, treat :math:`N(0,1)` as the name of this **standard reference
distribution**. We will develop the general normal family in the next section.

Its density is symmetric about zero. For reference, the density is

.. math::

   \phi(z)
   =
   \frac{1}{\sqrt{2\pi}}
   e^{-z^2/2},
   \qquad -\infty<z<\infty.

We will **not** calculate probabilities by integrating this function by hand.
Instead, we will use its cumulative probability function.

.. image:: ../_static/figures/continuous_distributions/figure_4_4_1_standard_normal_z_scale.png
   :width: 92%
   :align: center
   :alt: Standard normal reference curve with z-score landmarks

The horizontal scale is already measured in standard deviations. Therefore,
:math:`z=1` means one standard deviation above the mean and :math:`z=-2` means
two standard deviations below the mean.

Useful reference areas are

.. math::

   P(-1<Z<1)\approx0.6827,

.. math::

   P(-2<Z<2)\approx0.9545,

and

.. math::

   P(-3<Z<3)\approx0.9973.

These are useful landmarks for checking whether a probability calculation is
reasonable.

4.4.5 The Standard Normal CDF
------------------------------

Define the cumulative distribution function

.. math::

   \Phi(z)
   =
   P(Z\leq z).

Because :math:`Z` is continuous,

.. math::

   P(Z<z)=P(Z\leq z).

Thus,

.. math::

   \Phi(z)=P(Z<z).

Geometrically, :math:`\Phi(z)` is the area under the standard normal curve to
the **left** of :math:`z`.

The cumulative probability can also be written as

.. math::

   \Phi(z)
   =
   \int_{-\infty}^{z}\phi(t)\,dt.

This integral does not have a simple elementary antiderivative. Therefore,
probabilities are normally obtained from a **standard normal table** or
software.

4.4.6 Reading a Cumulative z-Table
-----------------------------------

This handout assumes a standard normal table whose entries are

.. math::

   \Phi(z)=P(Z<z).

Always check the heading of a table before using it because some tables use a
different convention.

Suppose we want

.. math::

   P(Z<1.48).

To locate :math:`z=1.48`:

1. use :math:`1.4` to locate the row;
2. use :math:`0.08` to locate the column;
3. read the entry at their intersection.

.. image:: ../_static/figures/continuous_distributions/figure_4_4_3_z_table_lookup.png
   :width: 94%
   :align: center
   :alt: Standard normal cumulative table lookup for z equals 1.48

The table gives

.. math::

   \Phi(1.48)=0.9306.

Therefore,

.. math::

   P(Z<1.48)=0.9306.

A negative value is read in the same way if the table includes negative
:math:`z`-scores. For example,

.. math::

   \Phi(-0.25)=0.4013,

so

.. math::

   P(Z<-0.25)=0.4013.

4.4.7 Three Basic Probability Patterns
---------------------------------------

Most standard-normal calculations reduce to three patterns.

.. image:: ../_static/figures/continuous_distributions/figure_4_4_2_standard_normal_probability_patterns.png
   :width: 98%
   :align: center
   :alt: Left tail right tail and interval probability patterns for the standard normal distribution

**Pattern 1: Left of a value**

.. math::

   P(Z<z)=\Phi(z).

This is read directly from a cumulative table.

**Pattern 2: Right of a value**

The total area is 1, so

.. math::

   P(Z>z)
   =
   1-\Phi(z).

**Pattern 3: Between two values**

For :math:`a<b`,

.. math::

   P(a<Z<b)
   =
   \Phi(b)-\Phi(a).

The same rules work whether :math:`a` and :math:`b` are positive, negative, or
on opposite sides of zero.

4.4.8 Example: Left-Tail Probability
-------------------------------------

Find

.. math::

   P(Z<1.48).

From the table,

.. math::

   \Phi(1.48)=0.9306.

Therefore,

.. math::

   P(Z<1.48)=0.9306.

Interpretation: about 93.06% of the standard normal area lies to the left of
:math:`z=1.48`.

4.4.9 Example: Right-Tail Probability
--------------------------------------

Find

.. math::

   P(Z>1.60).

The table gives

.. math::

   \Phi(1.60)=0.9452.

Use the complement:

.. math::

   P(Z>1.60)
   =
   1-P(Z<1.60)

.. math::

   =1-0.9452
   =0.0548.

A right-tail probability for a positive :math:`z` should be relatively small,
which is consistent with this answer.

Now consider

.. math::

   P(Z>-1.02).

From the table,

.. math::

   \Phi(-1.02)=0.1539.

Therefore,

.. math::

   P(Z>-1.02)
   =
   1-0.1539
   =
   0.8461.

4.4.10 Example: Probability Between Two z-Scores
-------------------------------------------------

Find

.. math::

   P(0.50<Z<1.57).

From the cumulative table,

.. math::

   \Phi(0.50)=0.6915

and

.. math::

   \Phi(1.57)=0.9418.

Therefore,

.. math::

   P(0.50<Z<1.57)
   =
   \Phi(1.57)-\Phi(0.50)

.. math::

   =0.9418-0.6915
   =0.2503.

The same procedure works if the interval crosses zero.

For example,

.. math::

   P(-2.55<Z<0.09)
   =
   \Phi(0.09)-\Phi(-2.55).

Using

.. math::

   \Phi(0.09)=0.5359

and

.. math::

   \Phi(-2.55)=0.0054,

we obtain

.. math::

   P(-2.55<Z<0.09)
   =
   0.5359-0.0054
   =0.5305.

4.4.11 Symmetry of the Standard Normal Curve
---------------------------------------------

The standard normal curve is symmetric about zero.

Therefore,

.. math::

   \Phi(-z)=1-\Phi(z).

Equivalently,

.. math::

   P(Z<-z)=P(Z>z).

For example,

.. math::

   P(Z<-1.08)=P(Z>1.08).

Symmetry is especially useful if a table contains only positive
:math:`z`-scores.

Another useful result is

.. math::

   P(-a<Z<a)
   =
   2\Phi(a)-1.

The probability outside a symmetric interval is

.. math::

   P(Z<-a\text{ or }Z>a)
   =
   2[1-\Phi(a)].

For example, since

.. math::

   \Phi(1.96)\approx0.9750,

we obtain

.. math::

   P(-1.96<Z<1.96)
   \approx
   0.9500

and

.. math::

   P(Z<-1.96\text{ or }Z>1.96)
   \approx
   0.0500.

These values will appear frequently in later statistical inference topics.

4.4.12 Very Large Positive or Negative z-Scores
------------------------------------------------

Some printed tables stop near :math:`z=3.5` or :math:`z=4.0`.

If :math:`z` is far to the right, then

.. math::

   \Phi(z)\approx1.

If :math:`z` is far to the left, then

.. math::

   \Phi(z)\approx0.

For example,

.. math::

   \Phi(4.16)\approx1.0000

when rounded to four decimal places.

Thus,

.. math::

   P(1.13<Z<4.16)
   \approx
   1.0000-0.8708
   =0.1292.

Likewise,

.. math::

   \Phi(-5.22)\approx0.0000.

These approximations reflect extremely small tail areas.

4.4.13 From a Probability Back to a z-Score
--------------------------------------------

Sometimes the probability is known and the corresponding :math:`z`-value is
unknown.

Suppose

.. math::

   \Phi(z)=0.9500.

Search the body of the cumulative table for a value near :math:`0.9500` and
read the corresponding row and column.

This gives approximately

.. math::

   z\approx1.645.

Similarly,

.. math::

   \Phi(z)=0.9750

corresponds to

.. math::

   z\approx1.96.

This reverse lookup is the basis for finding **critical values** and
**percentiles** later in the course.

4.4.14 Training with z-Scores Before Probability
-------------------------------------------------

Before using a probability table, be comfortable interpreting the standardized
scale itself.

Suppose a measurement has

.. math::

   \mu=50,
   \qquad
   \sigma=6.

Then:

.. list-table::
   :header-rows: 1
   :widths: 25 25 50

   * - :math:`x`
     - :math:`z`
     - Interpretation
   * - 50
     - 0
     - at the mean
   * - 56
     - 1
     - one standard deviation above the mean
   * - 44
     - -1
     - one standard deviation below the mean
   * - 62
     - 2
     - two standard deviations above the mean
   * - 38
     - -2
     - two standard deviations below the mean

For example,

.. math::

   z
   =
   \frac{62-50}{6}
   =2.

The value 62 is therefore two standard deviations above the mean.

This interpretation does not require a probability calculation.

4.4.15 A Reliable Workflow for Standard Normal Problems
--------------------------------------------------------

When working with :math:`Z`, use the following procedure.

1. **Identify the requested region.**

   Decide whether the probability is left of, right of, between, or outside
   specified :math:`z`-values.

2. **Sketch the region mentally or on paper.**

   This helps determine whether the answer should be small or large.

3. **Read cumulative probabilities.**

   Use

   .. math::

      \Phi(z)=P(Z<z).

4. **Apply the appropriate probability rule.**

   - left tail: :math:`\Phi(z)`;
   - right tail: :math:`1-\Phi(z)`;
   - interval: :math:`\Phi(b)-\Phi(a)`;
   - symmetric tails: use symmetry when helpful.

5. **Check the answer.**

   A probability must lie between 0 and 1 and should agree with the shaded
   region you intended to calculate.

4.4.16 Quick Practice
----------------------

Use a cumulative standard normal table or software.

**1.** Find

.. math::

   P(Z<-1.72).

**2.** Find

.. math::

   P(Z>1.63).

**3.** Find

.. math::

   P(-0.93<Z<0.55).

**4.** Find

.. math::

   P(Z<-1.96\text{ or }Z>1.96).

**5.** A value is :math:`1.5` standard deviations below its mean. What is its
:math:`z`-score?

**6.** A measurement has :math:`\mu=80` and :math:`\sigma=5`. Find the
:math:`z`-score of :math:`x=92`.

**Answers**

1.

   .. math::

      P(Z<-1.72)=0.0427.

2.

   .. math::

      P(Z>1.63)
      =
      1-0.9484
      =0.0516.

3.

   .. math::

      P(-0.93<Z<0.55)
      =
      0.7088-0.1762
      =0.5326.

4.

   .. math::

      P(Z<-1.96\text{ or }Z>1.96)
      \approx0.0500.

5.

   .. math::

      z=-1.5.

6.

   .. math::

      z
      =
      \frac{92-80}{5}
      =2.4.

4.4.17 Common Mistakes
-----------------------

**Mistake 1: Reversing the numerator**

Use

.. math::

   z=\frac{x-\mu}{\sigma},

not :math:`(\mu-x)/\sigma`.

**Mistake 2: Ignoring the sign of z**

A negative :math:`z` is below the mean. A positive :math:`z` is above the mean.

**Mistake 3: Treating a cumulative table as a right-tail table**

If the table gives :math:`\Phi(z)=P(Z<z)`, a right-tail probability requires

.. math::

   1-\Phi(z).

**Mistake 4: Adding instead of subtracting for an interval**

For :math:`a<b`,

.. math::

   P(a<Z<b)
   =
   \Phi(b)-\Phi(a).

**Mistake 5: Worrying about < versus <=**

Because :math:`Z` is continuous,

.. math::

   P(Z<z)=P(Z\leq z).

**Mistake 6: Assuming every z-score follows the standard normal distribution**

A :math:`z`-score is a standardized position. Additional distributional
information is required before standard normal probabilities can be attached to
an arbitrary standardized variable.

4.4.18 Summary
--------------

A :math:`z`-score converts distance from the mean into standard deviation units:

.. math::

   z=\frac{x-\mu}{\sigma}.

The standard normal reference variable has

.. math::

   Z\sim N(0,1),

with mean 0 and standard deviation 1.

Its cumulative distribution function is

.. math::

   \Phi(z)=P(Z<z).

The main probability rules are

.. math::

   P(Z<z)=\Phi(z),

.. math::

   P(Z>z)=1-\Phi(z),

and

.. math::

   P(a<Z<b)=\Phi(b)-\Phi(a).

Symmetry gives

.. math::

   \Phi(-z)=1-\Phi(z).

The key skill from this section is to become comfortable **thinking on the
z-scale** and translating shaded regions into cumulative probabilities.

In the next section, we will use this standard reference scale to work with the
general normal distribution.