z-scores

# z-scores

**Topic 4 · Thursday, January 14**

]

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# Where are we?

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

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

???

Still description. On Tuesday we summarized a whole distribution in two numbers. Today we use those two numbers to say where one score sits.

---

# Plan for today

.box-1.medium.sp-after-half[z-scores and locations]

.box-6.medium.sp-after-half[Transforming distributions]

.box-3.medium.sp-after-half[Standardized distributions]

???

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

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# z-scores and locations

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# Andy scored 19. Is that good?

???

Andy isn't a real student in the data, but real students did score 19. Ask: how would you say how typical or unusual this score is? It's only 19 out of 35, so is that bad? A raw score means little until we know where the other scores are.

---

# Measure the distance in SDs

.box-inv-1.small[Each SD is 3.90 points. Andy's 7.88 points are 7.88 / 3.90 = 2.02 SDs]

???

Four questions. What's typical? The mean. How far is Andy from it? The deviation. How much do scores vary overall? The SD. So how many SDs away is he?

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# A z-score counts SDs from the mean

$$
z = \frac{X - \mu}{\sigma} = \frac{19 - 11.12}{3.90} = 2.02
$$

How far the score is from the mean, in points. This is the deviation from Tuesday
]

The ruler. It says how big a step is in this distribution
]

.box-1.small[Treat the 619 students as a population: μ = 11.12 and σ = 3.90]

???

The z has no units. Points divided by points. That's why it will work across different tests later today.

---

# Sign: which side. Size: how far

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:center;"> Score X </th>
   <th style="text-align:center;"> Deviation X − μ </th>
   <th style="text-align:center;"> z </th>
   <th style="text-align:center;"> In words </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 19 </td>
   <td style="text-align:center;"> +7.88 </td>
   <td style="text-align:center;"> +2.02 </td>
   <td style="text-align:center;"> About two SDs above the mean </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 15 </td>
   <td style="text-align:center;"> +3.88 </td>
   <td style="text-align:center;"> +0.99 </td>
   <td style="text-align:center;"> About one SD above the mean </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 11 </td>
   <td style="text-align:center;"> −0.12 </td>
   <td style="text-align:center;"> −0.03 </td>
   <td style="text-align:center;"> Almost exactly at the mean </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> −6.12 </td>
   <td style="text-align:center;"> −1.57 </td>
   <td style="text-align:center;"> About one and a half SDs below the mean </td>
  </tr>
</tbody>
</table>

.box-inv-1.small[z = 0 is the mean. Most of these students sit between z = −2 and z = +2: 94% of them]

---

# From a z-score back to a score

$$
X = \mu + z\sigma = 11.12 + 2.02 \times 3.90 = 19.00
$$

???

If you round z to 2, you get 18.92, not 19. Keep two decimals along the way. The lab and the homework will say how much to round.

---

# A sample works the same way

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Score to z </th>
   <th style="text-align:center;"> z to score </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Population </td>
   <td style="text-align:center;"> $z = \dfrac{X - \mu}{\sigma}$ </td>
   <td style="text-align:center;"> $X = \mu + z\sigma$ </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sample </td>
   <td style="text-align:center;"> $z = \dfrac{X - M}{\hat{\sigma}}$ </td>
   <td style="text-align:center;"> $X = M + z\hat{\sigma}$ </td>
  </tr>
</tbody>
</table>

.box-inv-1.small[
Tuesday's five: M = 12.2 and σ̂ = 2.86. In this sample, the 16 has z = 1.33
]

???

Same idea, different symbols. Greek means population, and a Latin letter or a hat means an estimate. A z-score in a sample uses the estimates.

---

# Try it: Go there and back

.box-8.small.sp-after-half[For the math test: μ = 11.12 and σ = 3.90]

.box-inv-8.small[**1.** A student scored 9. What's the z-score?]

.box-inv-8.small[**2.** A student has z = +1.50. What's the score?]

.box-inv-8.small.sp-after-half[**3.** In Tuesday's sample, M = 12.2 and σ̂ = 2.86. Find the z of the 8.]

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

???

About 6 minutes, with calculators. For the vote, ask for hands on the sign of each answer first, then call out the value.

---

# Try it: Answer

.box-8.small[A negative z-score means below the mean. It doesn't mean a bad score]

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# Transforming distributions

---

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# A 19 means nothing by itself

.box-inv-6.small[It's only a label. We could relabel every score, and everyone would keep their place]

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# Subtract 10: the labels slide

.box-inv-6.small[Mean: 11.12 → 1.12. The SD stays at 3.90. Andy is still 2.02 SDs above the mean]

---

# Multiply by 10: the labels stretch

.box-inv-6.small[Mean: 11.12 → 111.2. SD: 3.90 → 39.0. Andy is still 2.02 SDs above the mean]

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# Do both: subtract, then multiply

.box-inv-6.small[Mean: 11.12 → 11.2. SD: 3.90 → 39.0. Andy is still 2.02 SDs above the mean]

---

# These only change the labels

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;"> Do this to every score </th>
   <th style="text-align:center;"> Mean </th>
   <th style="text-align:center;"> SD </th>
   <th style="text-align:center;"> Shape </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Add or subtract a constant </td>
   <td style="text-align:center;"> Moves by the constant </td>
   <td style="text-align:center;"> Stays the same </td>
   <td style="text-align:center;"> Stays the same </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Multiply or divide by a positive constant </td>
   <td style="text-align:center;"> Multiplied or divided </td>
   <td style="text-align:center;"> Multiplied or divided </td>
   <td style="text-align:center;"> Stays the same </td>
  </tr>
</tbody>
</table>

.box-6.small.sp-after-half[We call these "scale-invariant" transformations]

.box-inv-6.small[They keep the shape and the relative distances between students]

???

The SD ignores a shift because it's built from distances between scores and the mean. Both move together, so the distances don't change.

---

# Squaring changes the shape

.box-inv-6.small[Now 4 and 6 are 20 apart, but 18 and 20 are 76 apart]

???

Not every transformation is a relabeling. Both pairs of students were 2 points apart. After squaring, 4 and 6 become 16 and 36, while 18 and 20 become 324 and 400. So this isn't a relabeling. It changed who is close to whom.

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# Predict it: Two steps at once

.box-inv-8.small[
Take every math score. Subtract 11.12, the mean. Then divide by 3.90, the SD. What are the new mean and SD?
]

.options[
.option[
**A** Mean 11.12, SD 3.90
]
.option[
**B** Mean 0, SD 3.90
]
.option[
**C** Mean 0, SD 1
]
.option[
**D** Mean 1, SD 0
]
]

???

About 6 minutes. Vote, turn to a neighbor, vote again, then show the answer. Point back to the table: what does subtracting do, then what does dividing do?

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.wrong[
**A** Mean 11.12, SD 3.90
]
.option.wrong[
**B** Mean 0, SD 3.90
]
.option.correct[
**C** Mean 0, SD 1
]
.option.wrong[
**D** Mean 1, SD 0
]
]

.box-inv-8.small.sp-after-half[
Subtracting moves the mean from 11.12 to 0. The SD doesn't change
]

.box-8.small[
Dividing takes the SD from 3.90 to 1. The mean stays at 0
]

---

# Standardized distributions

---

---

# A z-score is the same kind of relabeling

$$
z = \frac{X - \mu}{\sigma}
$$

That constant is the mean
]

That constant is the standard deviation
]

.box-3.small[Do it to every score and you get a *standardized* distribution]

---

# Same picture, new axis: mean 0, SD 1

.box-inv-3.small[The z-scores have mean 0.00 and SD 1.00. Every student keeps their place]

???

The book draws this with a normal curve. Here it's the real data. Nothing moved: we printed a second ruler on the plot.

---

# z-scores don't make a shape normal

.box-inv-3.small[A skewed distribution stays skewed. Only the axis changes]

???

This is a common mistake. Standardizing moves the center and sets the spread. It can't touch the shape. Math scores look bell-shaped because they already were.

---

# We can add labels we like better

$$
\text{New score} = \text{new mean} + z \times \text{new SD}
$$

.box-3.small[Same student, same place. Nobody needs to see a negative score or a decimal]

???

Why not stop at z? People don't like being told their score is −0.54. So test makers go one step further. It's one more scale-invariant transformation.

---

# Score reports are tables like this one

<table class="deck-table tight">
 <thead>
  <tr>
   <th style="text-align:center;"> Math score </th>
   <th style="text-align:center;"> z </th>
   <th style="text-align:center;"> Mean 500, SD 100 </th>
   <th style="text-align:center;"> Mean 100, SD 15 </th>
   <th style="text-align:center;"> % who scored lower </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 3 </td>
   <td style="text-align:center;"> −2.08 </td>
   <td style="text-align:center;"> 292 </td>
   <td style="text-align:center;"> 69 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 7 </td>
   <td style="text-align:center;"> −1.06 </td>
   <td style="text-align:center;"> 394 </td>
   <td style="text-align:center;"> 84 </td>
   <td style="text-align:center;"> 11 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 11 </td>
   <td style="text-align:center;"> −0.03 </td>
   <td style="text-align:center;"> 497 </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> 43 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 15 </td>
   <td style="text-align:center;"> +0.99 </td>
   <td style="text-align:center;"> 599 </td>
   <td style="text-align:center;"> 115 </td>
   <td style="text-align:center;"> 83 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 19 </td>
   <td style="text-align:center;"> +2.02 </td>
   <td style="text-align:center;"> 702 </td>
   <td style="text-align:center;"> 130 </td>
   <td style="text-align:center;"> 97 </td>
  </tr>
</tbody>
</table>

.box-inv-3.small[The study's scaled math scores do this too: mean 0.00, SD 1.00 at baseline]

???

Tests like the SAT and GRE report a scaled score and a percentile. Each testing program picks its own scale, and the percentile comes from the people who took it.

The study's scaled scores come from a measurement model, not from this simple formula. The idea of reporting in SDs is the same. Don't go further than that unless someone asks.

---

# A 19 in math beats a 19 in Hindi

.box-inv-3.small[Percent correct says Hindi: 63% against 54%. z-scores say math: 2.02 against 1.25]

???

Ask first: which test did Andy do better on? The tests differ in length, average and spread, so raw scores can't be compared. z-scores have no units, so they can. 2 real students scored 19 on both tests.

Andy beat 97% of students in math and 86% in Hindi.

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# The blueprint starts here

???

Click to draw a few times. Panel 1 is the population, a smooth curve with our test's mean and SD. Panel 2 is one score and its z. Panel 3 stacks the z-scores. This is step 1 of the blueprint, and every test later in the course has these three panels. Ignore the cutoffs and the p for now. They arrive with hypothesis testing.

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# Play it: z-scores

.box-8.small.sp-after-half[Answer with a neighbor, as a question. Nothing is scored]

???

About 6 minutes. Read one clue at a time, in order. No calculators. Take answers by hands or by calling out.

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# Play it: Answers

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# What you can do now

.box-inv-8.small[Turn a score into a z-score, and a z-score back into a score]

.box-inv-8.small[Say what the sign and the size of a z-score mean]

.box-inv-8.small[Say what adding, multiplying and standardizing do to a distribution]

.box-inv-8.small[Compare scores from two different tests]

???

Topic 5 on Tuesday is probability. It connects a z-score to how likely a score is.

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# Next steps

Read Chapter 6
]

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]

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