Central tendency and variability

# Central tendency and variability

**Topic 3 · Tuesday, January 12**

]

---

# Where are we?

.box-inv-1.small[Summarize data]

.box-inv-1.small[Visualize data]

???

Same map as last week. Today we summarize a whole distribution in two numbers: where it sits and how spread out it is.

---

# Plan for today

.box-1.medium.sp-after-half[What's typical?]

.box-6.medium.sp-after-half[Shape and spread]

.box-3.medium.sp-after-half[The standard deviation]

???

Three blocks of about 20 minutes. Each ends with students doing something: predict, try, predict.

---

# What's typical?

---

---

# What value best describes this group?

???

Ask: which single number would you report? 12, because it's the tallest bar? Why not 11? Or 11.5?

---

# Three ways to name a "typical" value

The most frequent value

Find the value that occurs most often
]

The midpoint of a distribution

Rank the scores and take the middle one. With an even number, average the middle two
]

The average, or the balance point

Add up all the scores, then divide by how many there are
]

---

# The mean: one idea, two symbols

$$
\mu = \frac{\sum X}{N} \qquad\qquad M = \frac{\sum X}{n}
$$

**μ** is the mean of everyone we care about. There are **N** of them
]

**M** is the mean of the **n** people we observed. It estimates μ
]

.box-1.small[Greek means population. A Latin letter or a hat means an estimate]

???

This rule comes back all quarter. Students should be able to label any symbol as a parameter or an estimate.

---

# Here, all three land close together

---

# Predict it: One perfect score

.box-inv-8.small[
Five students scored 8, 12, 12, 13, 13. Their mean is 11.6 and their median is 12. Suppose one 13 becomes a perfect 35. What happens?
]

.options[
.option[
**A** Only the mean changes
]
.option[
**B** Only the median changes
]
.option[
**C** Both change
]
.option[
**D** Neither changes
]
]

???

About 6 minutes. Vote, turn to a neighbor, vote again, then show the answer.

Andy: create this poll in Poll Everywhere, then add the `.poll-bar` button from EDUC 265's 06-01 with its presenter id.

---

# Predict it: Answer

.options[
.option.correct[
**A** Only the mean changes
]
.option.wrong[
**B** Only the median changes
]
.option.wrong[
**C** Both change
]
.option.wrong[
**D** Neither changes
]
]

.box-inv-8.small.sp-after-half[
The mean goes from 11.6 to 16.0. The median stays at 12
]

.box-8.small[
The mean uses every score's value. The median only uses their order
]

---

# Shape and spread

---

---

# Skew is named after the tail

.pull-left-3[
.dog[![Cartoon dog lying down, with a long tail stretching to the left](img/dog-tail-left.png)]
]

.pull-right-3[
.dog[![Cartoon dog lying down, with a long tail stretching to the right](img/dog-tail-right.png)]
]

.box-inv-6.small[Our math scores are almost symmetric, so all three agreed]

---

# In a skewed distribution, they pull apart

.box-inv-6.small[The mean gets pulled toward the tail. That's the perfect score you just saw]

---

# So which one should you report?

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;"> Measure </th>
   <th style="text-align:left;"> Works well for </th>
   <th style="text-align:left;"> Can mislead with </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Mean </td>
   <td style="text-align:left;"> Interval or ratio scales, when you know every score </td>
   <td style="text-align:left;"> Extreme scores, skewed distributions, open-ended categories </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Median </td>
   <td style="text-align:left;"> Extreme scores, skewed distributions, open-ended categories, ordinal scales </td>
   <td style="text-align:left;"> Nominal scales </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Mode </td>
   <td style="text-align:left;"> Nominal scales, discrete variables, describing shape </td>
   <td style="text-align:left;"> Interval or ratio data, unless it comes with the mean or median </td>
  </tr>
</tbody>
</table>

---

# The range: how far the scores stretch

.box-6.medium.sp-after-half[Range = highest score − lowest score]

.box-inv-6.small.sp-after-half[
In our data: 26 − 0 = 26 points
]

.box-inv-6.small[
It's easy to find, but only two scores set it
]

???

The book also defines the range with real limits for continuous variables. That detail isn't worth class time.

---

# The interquartile range: the middle 50%

.box-inv-6.small[IQR = 14 − 9 = 5 points. One extreme student can't move it]

---

# Try it: Five real students

.box-inv-8.medium.sp-after-half[Scores: 8, 13, 12, 12, 16]

.box-8.small.sp-after-half[Find the mean, the median and the range]

.box-inv-8.small[Work alone for two minutes. Compare with a neighbor. Then we vote]

???

About 6 minutes, with calculators. Students who finish early can write down why the mean and median differ.

---

# Try it: Answer

---

# The standard deviation

---

---

# How far is a typical score from μ?

.box-inv-3.small[Say these five students are everyone we care about, so μ = 12.2]

<table class="deck-table tight">
 <thead>
  <tr>
   <th style="text-align:center;"> Score X </th>
   <th style="text-align:center;"> Deviation X − μ </th>
   <th style="text-align:center;"> Squared deviation </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 8 </td>
   <td style="text-align:center;"> −4.2 </td>
   <td style="text-align:center;"> 17.64 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 13 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.64 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 12 </td>
   <td style="text-align:center;"> −0.2 </td>
   <td style="text-align:center;"> 0.04 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 12 </td>
   <td style="text-align:center;"> −0.2 </td>
   <td style="text-align:center;"> 0.04 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 16 </td>
   <td style="text-align:center;"> 3.8 </td>
   <td style="text-align:center;"> 14.44 </td>
  </tr>
</tbody>
</table>

.box-3.small[Add up the last column to get the sum of squares: SS = 32.8]

???

Why square? Deviations from the mean always add up to zero, so their plain average tells us nothing.

---

# Four steps to the standard deviation

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;"> Step </th>
   <th style="text-align:left;"> In symbols </th>
   <th style="text-align:left;"> Our five students </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1. Find each deviation </td>
   <td style="text-align:left;"> $X - \mu$ </td>
   <td style="text-align:left;"> The table we just built </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2. Square them and add them up </td>
   <td style="text-align:left;"> $SS = \sum (X - \mu)^2$ </td>
   <td style="text-align:left;"> SS = 32.8 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3. Average them: the variance </td>
   <td style="text-align:left;"> $\sigma^2 = SS / N$ </td>
   <td style="text-align:left;"> 32.8 / 5 = 6.56 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4. Take the root: the SD </td>
   <td style="text-align:left;"> $\sigma = \sqrt{\sigma^2}$ </td>
   <td style="text-align:left;"> √6.56 = 2.56 </td>
  </tr>
</tbody>
</table>

.box-3.small[Variance is the average squared distance. The SD puts it back in points]

---

# We rarely see the whole population

.box-inv-3.small[A small sample rarely catches the extreme scores, so it tends to look less spread out than its population]

---

# Predict it: Treat the 619 students as a population

.box-inv-8.small.sp-after-half[Their variance is σ² = 15.22. Draw 5,000 samples of 5. In each one, compute SS / n]

.box-8.small.sp-after-half[Across all 5,000 samples, what does SS / n average out to?]

.options[
.option[
**A** Less than 15.22
]
.option[
**B** About 15.22
]
.option[
**C** More than 15.22
]
]

???

About 6 minutes. Hands are fine here. Ask students to say why before showing the next slide.

---

# Predict it: SS / n falls short on average

.box-inv-8.small[SS / n tends to be too small, though 28% of samples still overshoot. Dividing by n − 1 lands on target on average]

---

---

# Our estimate of the population variance

$$
\hat{\sigma}^2 = \frac{SS}{n - 1} \qquad\qquad \hat{\sigma} = \sqrt{\hat{\sigma}^2}
$$

.pull-left[
.box-inv-3.small[
The hat says this is an estimate. n − 1 is called the *degrees of freedom*
]
]

.pull-right[
.box-inv-3.small[
SS / n is still the sample's own variance. It just isn't our best guess about the population
]
]

???

The variance estimate is unbiased. Its square root runs a little low: here it averages 3.61 against a true SD of 3.90. That's worth saying only if someone asks.

---

# Our five students, now as a sample

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Population </th>
   <th style="text-align:center;"> Sample </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> How many </td>
   <td style="text-align:center;"> N </td>
   <td style="text-align:center;"> n = 5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Mean </td>
   <td style="text-align:center;"> μ </td>
   <td style="text-align:center;"> M = 12.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Variance </td>
   <td style="text-align:center;"> σ² = SS / N </td>
   <td style="text-align:center;"> σ̂² = 32.8 / 4 = 8.20 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Standard deviation </td>
   <td style="text-align:center;"> σ </td>
   <td style="text-align:center;"> σ̂ = 2.86 </td>
  </tr>
</tbody>
</table>

.box-3.small[With all 619 students, n − 1 hardly matters: σ̂ = 3.90, against 3.90 when dividing by n]

---

# What you can do now

.box-inv-8.small[Find and choose among the mean, median and mode]

.box-inv-8.small[Say how skew pulls the mean toward the tail]

.box-inv-8.small[Compute SS, the variance and the SD by hand]

.box-inv-8.small[Label a symbol as a parameter or an estimate, and say why we divide by n − 1]

---

# Next steps

Read Chapter 5
]

Bring a calculator and a laptop. It includes a 15-minute individual check
]

.box-3.small[Mastery and Problem Set for Chapter 5 are due Sunday at 6 PM]

---

# Bonus: A faster way to compute SS

$$
SS = \sum (X - \mu)^2 = \sum X^2 - 2\mu \sum X + N\mu^2
$$

$$
SS = \sum X^2 - \frac{(\sum X)^2}{N}
$$

.box-inv-3.small[This needs no deviations, so it's quicker with a calculator]

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