# Finding outliers with standard deviation and regression#

To reproduce this finding from the Dallas Morning News, we'll need to use standard deviation and regression to identify schools that performed suspiciously well in certain standardized tests.

## Finding suspicious behavior by tracking down outliers#

```import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.api as sm
import numpy as np
from statsmodels.sandbox.regression.predstd import wls_prediction_std

pd.set_option("display.max_rows", 200)
pd.set_option("display.max_columns", 200)
```

We'll start by opening up our dataset - standardized test performance at each school, for fourth graders in 2004.

```df = pd.read_csv("data/cfy04e4.dat", usecols=['r_all_rs', 'CNAME', 'CAMPUS'])
```
CNAME_fourth r_all_rs_fourth
CAMPUS
1902103 CAYUGA EL 2392.0
1903101 ELKHART EL 2263.0
1904102 FRANKSTON EL 2242.0
1906102 NECHES EL 2218.0
1907110 STORY EL 2200.0

That dataset had a lot of columns, but we're only interested in their reading scores.

How can we find out who did suspiciously well? We should probably start with figuring out which schools performed normally. Statistically speaking, "normal performance" is probably the median (one of the three types of averages).

```df.r_all_rs_fourth.median()
```
`2226.0`

The median is the 50% mark of the score: half of the schools did better than 2226, and half of the schools did worse.

As a school's score gets further and further from the median, the school is going further and further from being an "average school." If a schools get a lot lot lot of points, it should probably be looked at. But how many extra points is enough to make the school suspicious?

```df.r_all_rs_fourth.describe()
```
```count    3706.000000
mean     2227.245008
std        68.598894
min      1951.000000
25%      2181.000000
50%      2226.000000
75%      2273.000000
max      2509.000000
Name: r_all_rs_fourth, dtype: float64```

If 2226 points is the average, 2273 puts a school at the 75% mark. 75% still doesn't seem very suspicious, though, as that's still one out of every four schools. We want something higher! 95%? 99.7%?

```df.r_all_rs_fourth.hist(bins=20)
```
`<matplotlib.axes._subplots.AxesSubplot at 0x11cb026d8>`

We want those very few datapoints on the far right!

Instead of picking an arbitrary number or percentile, we're going to start using a statistical measure called the standard deviation. It's a measurement of how spread out the data is (`std` in the list above). To explain how unusual a data point is, you can say "it's 1.5 standard deviations from the mean" or "2.75 standard deviations from the mean."

```df['std_dev_rs_fourth'] = (df.r_all_rs_fourth - df.r_all_rs_fourth.mean()) / df.r_all_rs_fourth.std()
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth
CAMPUS
1902103 CAYUGA EL 2392.0 2.401715
1903101 ELKHART EL 2263.0 0.521218
1904102 FRANKSTON EL 2242.0 0.215091
1906102 NECHES EL 2218.0 -0.134769
1907110 STORY EL 2200.0 -0.397164

Now every school has a new column that explains how many standard deviations away from the average score it is. We can easily look at the top and bottom performers!

```df.sort_values(by='std_dev_rs_fourth').head(20)
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth
CAMPUS
167903001 STAR SCHOOL 1951.0 -4.026960
101912386 DIVERSITY ROOTS 1958.0 -3.924918
101912140 DOGAN EL 1972.0 -3.720833
101912114 BRAEBURN EL 1972.0 -3.720833
57817101 FOCUS LEARNING 1973.0 -3.706255
57830002 INSPIRED VISION 1979.0 -3.618790
18903001 MORGAN SCHOOL 1999.0 -3.327240
101912129 CLINTON PARK EL 2005.0 -3.239775
57825012 DESTINY HIGH SC 2013.0 -3.123155
105802041 SAN MARCOS PREP 2025.0 -2.948225
123801101 ACADEMY OF BEAU 2033.0 -2.831606
14803101 TEMPLE EDUCATIO 2033.0 -2.831606
66005101 RAMIREZ EL 2033.0 -2.831606
101851001 HOUSTON ALTERNA 2037.0 -2.773296
84902109 SAN JACINTO EL 2042.0 -2.700408
57830001 INSPIRED VISION 2043.0 -2.685831
152901115 JACKSON EL 2044.0 -2.671253
57905161 JOHN IRELAND EL 2044.0 -2.671253
101912155 FRANKLIN EL 2045.0 -2.656676
```df.sort_values(by='std_dev_rs_fourth', ascending=False).head(20)
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth
CAMPUS
131001103 SARITA EL 2509.0 4.107282
101912296 T H ROGERS EL 2480.0 3.684535
57905115 HARRELL BUDD EL 2470.0 3.538760
180901101 MIMI FARLEY ELE 2448.0 3.218055
68901118 EL MAGNET AT RE 2432.0 2.984815
57909122 WALNUT GLEN ACA 2430.0 2.955660
57916123 PRAIRIE CREEK E 2426.0 2.897350
57916128 BRENTFIELD EL 2425.0 2.882772
199901110 GRACE HARTMAN E 2419.0 2.795307
101920107 HUNTERS CREEK E 2418.0 2.780730
227909106 BRIDGE POINT EL 2414.0 2.722420
57911104 UNIVERSITY PARK 2413.0 2.707842
148905101 DARROUZETT EL 2412.0 2.693265
43910122 MATHEWS EL 2410.0 2.664110
57911103 HYER ELEMENTARY 2408.0 2.634955
43907108 VALLEY CREEK EL 2403.0 2.562067
57922106 WILSON EL 2402.0 2.547490
170902123 BUCKALEW EL 2401.0 2.532912

# But let's not get ahead of ourselves!#

The thing is, though: some schools are just going to be better. Getting a good score doesn't mean a school is cheating, it just means, well, that they got a good score!

We could use this list of standard deviations to investigate each and every school that did well, seeing if it makes sense that they did so well. That'd be good reporting! But it might also be a waste of time, as there wasn't anything "unusual" about these schools, they just... did well.

Now we need to ask ourselves when could a test score be suspicious? The Dallas Morning News realized they could look at scores across years at the same school - we were looking at fourth graders just now, but how did those students perform when they were in third grade? Did they just do average, and now suddenly they're geniuses? Suspicious!

Since we were working with 2004's fourth grader data, we'll now combine it with 2003's third-grader data.

```third_graders = pd.read_csv("data/cfy03e3.dat", usecols=['CAMPUS', 'r_all_rs'])
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third
CAMPUS
1902103 CAYUGA EL 2392.0 2.401715 2330.0
1903101 ELKHART EL 2263.0 0.521218 2285.0
1904102 FRANKSTON EL 2242.0 0.215091 2299.0
1906102 NECHES EL 2218.0 -0.134769 2236.0
1907110 STORY EL 2200.0 -0.397164 2202.0

# Graphing for research#

One of the things we could do is plot the third grade scores as compared to the fourth grade scores, and see if anything stands out.

```fig, ax = plt.subplots(figsize=(4,4))

ax.set_xlim(2000, 2500)
ax.set_ylim(1900, 2500)
ax.set_facecolor('lightgrey')
ax.grid(True, color='white')
ax.set_axisbelow(True)

sns.regplot('r_all_rs_third',
'r_all_rs_fourth',
data=merged,
marker='.',
line_kws={"color": "black", "linewidth": 1},
scatter_kws={"color": "grey"})

highlight = merged.loc[57905115]
plt.plot(highlight.r_all_rs_third, highlight.r_all_rs_fourth, 'ro')
```
`[<matplotlib.lines.Line2D at 0x11cc5e668>]`

This is one of the graphics from the Dallas Morning News Piece, and we've highlighted one of the suspicious schools, Harrell Budd Elementary.

```merged.loc[57905115]
```
```CNAME_fourth         HARRELL BUDD EL
r_all_rs_fourth                 2470
std_dev_rs_fourth            3.53876
r_all_rs_third                  2140
Name: 57905115, dtype: object```

You'd expect a school to perform similarly between third and fourth grade, but somehow this one school jumped from about 2150 points to 2500 points in one year! Seems suspicious.

But why is that suspicious? It's because we don't expect a school to make a jump of 350 points in one year, going from average to a high performer.

But what do we expect a school to do? I'm not sure, but just like The Dallas Morning News we can surely ask statistics for help!

# Using a regression to predict fourth-grade scores#

The Dallas Morning News decided to run a regression, which is a way of predicting how two different variables interact. In this case, we want to see the relationship between a third grade score and a fourth grade score.

First we'll need to get rid of missing data, because regressions hate hate hate missing data.

```print("Before dropping missing data", merged.shape)
merged = merged.dropna()
print("After dropping missing data", merged.shape)
```
```Before dropping missing data (3501, 4)
After dropping missing data (3501, 4)
```

```import statsmodels.formula.api as smf

model = smf.ols("r_all_rs_fourth ~ r_all_rs_third", data=merged)

results = model.fit()
results.summary()
```
Dep. Variable: R-squared: r_all_rs_fourth 0.647 OLS 0.647 Least Squares 6410. Tue, 05 Nov 2019 0.00 07:38:00 -17961. 3501 3.593e+04 3499 3.594e+04 1 nonrobust
coef std err t P>|t| [0.025 0.975] 632.6929 19.936 31.736 0.000 593.606 671.780 0.7094 0.009 80.061 0.000 0.692 0.727
 Omnibus: Durbin-Watson: 263.054 1.897 0 1312.9 0.147 8.08e-286 5.986 64900

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 6.49e+04. This might indicate that there are
strong multicollinearity or other numerical problems.

What's this all mean? It doesn't matter! What matters is that if we know what third-grade score a school got, we can predict its fourth-grade score.

```# What should these schools have gotten?
sample = pd.DataFrame({ 'r_all_rs_third': [2140, 2200, 2500] })
results.predict(sample)
```
```0    2150.828391
1    2193.392939
2    2406.215677
dtype: float64```
```merged['predicted_fourth'] = results.predict()
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third predicted_fourth
CAMPUS
1902103 CAYUGA EL 2392.0 2.401715 2330.0 2285.616126
1903101 ELKHART EL 2263.0 0.521218 2285.0 2253.692715
1904102 FRANKSTON EL 2242.0 0.215091 2299.0 2263.624443
1906102 NECHES EL 2218.0 -0.134769 2236.0 2218.931668
1907110 STORY EL 2200.0 -0.397164 2202.0 2194.811757

## Using standard deviations with regression error#

Notice how there's a difference between the actual fourth-grade score and the predicted fourth-grade score. This is called the error or residual. The bigger the error, the bigger the difference between what was expected and what actually happened.

Remember how we were suspicious of that one school because it performed normally, but then performed really well? A school like that is going to have a really big error!

To calculate what a "big error" is, we're going to use our old friend standard deviation. Before we used standard deviation to see how far a schools' score was from the average score. This time we're going to use standard deviation to see how far the school's error is from the average error!

```# Just trust me, this is how you do it
merged['error_std_dev'] = results.resid / np.sqrt(results.mse_resid)
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third predicted_fourth error_std_dev
CAMPUS
1902103 CAYUGA EL 2392.0 2.401715 2330.0 2285.616126 2.600187
1903101 ELKHART EL 2263.0 0.521218 2285.0 2253.692715 0.227484
1904102 FRANKSTON EL 2242.0 0.215091 2299.0 2263.624443 -0.528535
1906102 NECHES EL 2218.0 -0.134769 2236.0 2218.931668 -0.022771
1907110 STORY EL 2200.0 -0.397164 2202.0 2194.811757 0.126809

The more standard deviations away from the mean a school's error is, the more suspicious its fourth-grade performance is.

```merged.sort_values(by='error_std_dev', ascending=False).head(10)
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third predicted_fourth error_std_dev
CAMPUS
57905115 HARRELL BUDD EL 2470.0 3.538760 2140.0 2150.828391 7.801048
123803101 TEKOA ACADEMY O 2313.0 1.250093 2021.0 2066.408705 6.027073
15803101 HIGGS/CARTER/KI 2349.0 1.774883 2097.0 2120.323799 5.589200
101912172 HENDERSON N EL 2324.0 1.410445 2093.0 2117.486162 5.047518
180901101 MIMI FARLEY ELE 2448.0 3.218055 2294.0 2260.077397 4.593119
57825001 PINNACLE SCHOOL 2274.0 0.681571 2068.0 2099.750934 4.258917
131001103 SARITA EL 2509.0 4.107282 2424.0 2352.300584 3.829976
101823101 CHILDREN FIRST 2252.0 0.360866 2066.0 2098.332116 3.755881
61911111 PRAIRIE VIEW EL 2389.0 2.357983 2261.0 2236.666896 3.723257
57905185 WILLIAM B MILLE 2295.0 0.987698 2129.0 2143.024891 3.714507

# Reproducing the story#

From The Dallas Morning News:

"In statistician's lingo, these schools had at least one average score that was more than three standard deviations away from what would be predicted based on their scores in other grades or on other tests

While we've been talking about schools with a major increase between the two years, we're also interested in schools with a major drop. That could indicate cheating in 2003 and a return to "real" testing in 2004.

Let's check out all of our suspicious schools according to the three standard deviations test they performed.

```merged[merged.error_std_dev.abs() > 3]
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third predicted_fourth error_std_dev
CAMPUS
15803101 HIGGS/CARTER/KI 2349.0 1.774883 2097.0 2120.323799 5.589200
15819001 SHEKINAH RADIAN 1976.0 -3.662523 2149.0 2157.213074 -4.429128
19901102 DEKALB EL 2356.0 1.876925 2223.0 2209.709349 3.575570
22004101 TERLINGUA EL 2055.0 -2.510901 2178.0 2177.785938 -3.001078
25904101 BLANKET EL 2082.0 -2.117308 2225.0 2211.128167 -3.156092
47905001 SIDNEY SCHOOL 2208.0 -0.280544 2024.0 2068.536933 3.408693
50901101 EVANT EL 2169.0 -0.849066 2345.0 2296.257263 -3.110365
57817101 FOCUS LEARNING 1973.0 -3.706255 2079.0 2107.554435 -3.288719
57825001 PINNACLE SCHOOL 2274.0 0.681571 2068.0 2099.750934 4.258917
57825012 DESTINY HIGH SC 2013.0 -3.123155 2133.0 2145.862528 -3.247366
57905115 HARRELL BUDD EL 2470.0 3.538760 2140.0 2150.828391 7.801048
57905185 WILLIAM B MILLE 2295.0 0.987698 2129.0 2143.024891 3.714507
61911111 PRAIRIE VIEW EL 2389.0 2.357983 2261.0 2236.666896 3.723257
66005101 RAMIREZ EL 2033.0 -2.831606 2195.0 2189.845893 -3.833556
90905101 GRANDVIEW-HOPKI 2149.0 -1.140616 2346.0 2296.966672 -3.616535
101820001 BENJI'S SPECIAL 2131.0 -1.403011 2290.0 2257.239761 -3.085495
101823101 CHILDREN FIRST 2252.0 0.360866 2066.0 2098.332116 3.755881
101840101 TWO DIMENSIONS 2076.0 -2.204773 2275.0 2246.598624 -4.169694
101909104 LAKEWOOD ELEMEN 2275.0 0.696148 2104.0 2125.289663 3.659152
101912108 BASTIAN EL 2260.0 0.477486 2105.0 2125.999072 3.275190
101912134 CRAWFORD EL 2056.0 -2.496323 2240.0 2221.769304 -4.051658
101912140 DOGAN EL 1972.0 -3.720833 2150.0 2157.922483 -4.544233
101912172 HENDERSON N EL 2324.0 1.410445 2093.0 2117.486162 5.047518
101912217 PECK EL 2105.0 -1.782026 2265.0 2239.504532 -3.287499
103902001 HARTLEY SCHOOL 2288.0 0.885656 2159.0 2164.307165 3.023244
105802041 SAN MARCOS PREP 2025.0 -2.948225 2245.0 2225.316350 -4.896042
108912110 JOSE DE ESCANDO 2166.0 -0.892799 2360.0 2306.898400 -3.443775
123803101 TEKOA ACADEMY O 2313.0 1.250093 2021.0 2066.408705 6.027073
123907105 LEE EL 2292.0 0.943966 2165.0 2168.563620 3.016976
131001103 SARITA EL 2509.0 4.107282 2424.0 2352.300584 3.829976
139908101 ROXTON EL 2130.0 -1.417589 2388.0 2326.761855 -4.809164
180901101 MIMI FARLEY ELE 2448.0 3.218055 2294.0 2260.077397 4.593119
226903111 FT CONCHO EL 2196.0 -0.455474 2386.0 2325.343037 -3.161344
240901124 LIGARDE EL 2096.0 -1.913223 2239.0 2221.059895 -3.056657

But then they level things up a bit:

Using a stricter standard - four standard deviations from predictions - 41 schools have suspect scores

```merged[merged.error_std_dev.abs() > 4]
```
CNAME_fourth r_all_rs_fourth std_dev_rs_fourth r_all_rs_third predicted_fourth error_std_dev
CAMPUS
15803101 HIGGS/CARTER/KI 2349.0 1.774883 2097.0 2120.323799 5.589200
15819001 SHEKINAH RADIAN 1976.0 -3.662523 2149.0 2157.213074 -4.429128
57825001 PINNACLE SCHOOL 2274.0 0.681571 2068.0 2099.750934 4.258917
57905115 HARRELL BUDD EL 2470.0 3.538760 2140.0 2150.828391 7.801048
101840101 TWO DIMENSIONS 2076.0 -2.204773 2275.0 2246.598624 -4.169694
101912134 CRAWFORD EL 2056.0 -2.496323 2240.0 2221.769304 -4.051658
101912140 DOGAN EL 1972.0 -3.720833 2150.0 2157.922483 -4.544233
101912172 HENDERSON N EL 2324.0 1.410445 2093.0 2117.486162 5.047518
105802041 SAN MARCOS PREP 2025.0 -2.948225 2245.0 2225.316350 -4.896042
123803101 TEKOA ACADEMY O 2313.0 1.250093 2021.0 2066.408705 6.027073
139908101 ROXTON EL 2130.0 -1.417589 2388.0 2326.761855 -4.809164
180901101 MIMI FARLEY ELE 2448.0 3.218055 2294.0 2260.077397 4.593119

Our dataset isn't as thorough as theirs - we're only looking at one combination of tests - but it's the same idea.

# Finding other suspicious scores#

We might assume a school that does well in reading probably also does well in math.

What if they did well in one, but not the other? While the school might just have a strong department in one particular field, such discrepancies could be worth investigating.

```df = pd.read_csv("data/cfy04e5.dat", usecols=['CAMPUS', 'CNAME', 'm_all_rs', 'r_all_rs'])
```
CNAME_fifth r_all_rs_fifth m_all_rs_fifth
CAMPUS
1902103 CAYUGA EL 2308.0 2317.0
1903101 ELKHART EL 2193.0 2153.0
1904102 FRANKSTON EL 2288.0 2256.0
1906102 NECHES EL 2298.0 2312.0
1907110 STORY EL 2218.0 2269.0

# Building the graphic#

While it isn't necessary, reproducing the graphics is always fun.

```fig, ax = plt.subplots(figsize=(4,4))

ax.set_xlim(1900, 2500)
ax.set_ylim(1800, 2750)
ax.set_facecolor('lightgrey')
ax.grid(True, color='white')
ax.set_axisbelow(True)

sns.regplot('r_all_rs_fifth',
'm_all_rs_fifth',
data=df,
marker='.',
line_kws={"color": "black", "linewidth": 1},
scatter_kws={"color": "grey"})

highlight = df.loc[101912236]
plt.plot(highlight.r_all_rs_fifth, highlight.m_all_rs_fifth, 'ro')
```
`[<matplotlib.lines.Line2D at 0x11d58e400>]`

# Running the regression#

We can't be exactly sure of the relationship between math and reading scores - it's a lot of schools! - so we'll run a regression to figure out how the two scores typically interact.

```print("Before dropping missing data", df.shape)
df = df.dropna()
print("After dropping missing data", df.shape)
```
```Before dropping missing data (3571, 3)
After dropping missing data (3452, 3)
```
```import statsmodels.formula.api as smf

model = smf.ols("m_all_rs_fifth ~ r_all_rs_fifth", data=df)

results = model.fit()
results.summary()
```
Dep. Variable: R-squared: m_all_rs_fifth 0.735 OLS 0.735 Least Squares 9570. Tue, 05 Nov 2019 0.00 07:56:45 -18443. 3452 3.689e+04 3450 3.690e+04 1 nonrobust
coef std err t P>|t| [0.025 0.975] 136.3779 21.321 6.397 0.000 94.576 178.180 0.9462 0.010 97.826 0.000 0.927 0.965
 Omnibus: Durbin-Watson: 210.369 1.687 0 858.872 0.127 3.15e-187 5.43 54600

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 5.46e+04. This might indicate that there are
strong multicollinearity or other numerical problems.

And now, just like last time, we calculate how many standard deviations away the actual score was from the predicted score. Large number of standard deviations away means a school is worth a look!

```df['error_std_dev'] = results.resid / np.sqrt(results.mse_resid)

df[df.error_std_dev.abs() > 3].sort_values(by='error_std_dev', ascending=False)
```
CNAME_fifth r_all_rs_fifth m_all_rs_fifth error_std_dev
CAMPUS
101912236 SANDERSON EL 2235.0 2696.0 8.790097
109908101 MALONE EL 2104.0 2355.0 4.501120
101912113 BOWIE EL 2265.0 2498.0 4.316536
188901130 SUNRISE EL 2311.0 2517.0 3.831884
240901120 J C MARTIN JR E 2212.0 2417.0 3.706908
40901102 MORTON EL 2215.0 2417.0 3.650814
72909001 LINGLEVILLE SCH 2148.0 2017.0 -3.000714
233903001 COMSTOCK SCHOOL 2241.0 2100.0 -3.099482
178904152 BARNES EL 2233.0 2092.0 -3.107985
139911102 AARON PARKER EL 2126.0 1990.0 -3.122899
235904101 NURSERY EL 2314.0 2167.0 -3.140463
105905101 BOWEN INT 2313.0 2166.0 -3.141526
243801001 BRIGHT IDEAS CH 2211.0 2066.0 -3.210409
50909001 JONESBORO SCHOO 2127.0 1981.0 -3.319444
112906101 NORTH HOPKINS E 2204.0 2052.0 -3.356173
103902001 HARTLEY SCHOOL 2293.0 2130.0 -3.478953
227817101 CEDARS INTERNAT 2167.0 2006.0 -3.573343
21803001 BRAZOS SCHOOL F 2169.0 2007.0 -3.590978
148903001 HIGGINS SCHOOL 2195.0 2028.0 -3.662149
101805101 GIRLS & BOYS PR 2443.0 2262.0 -3.675228
70801001 WAXAHACHIE FAIT 2084.0 1912.0 -3.878922
19914101 LEARY EL 2296.0 2110.0 -3.930262
105802041 SAN MARCOS PREP 2203.0 1968.0 -4.997376

Wow, look at that! Sanderson Elementary looks like they either have a really exceptional math program or something suspicious is going on.

# Review#

First, we learned about using standard deviation as a measurement of how unusual a measurement in a data point might be. Data points that fall many standard deviations from the mean - either above or below - might be worth investigating as bad data or from other suspicious angles (cheating schools, in this case).

Then we learned how a linear regression can determine the relationship between two numbers. In this case, it was how third-grade scores relate to fourth-grade scores, and then how math and reading scores relate to one another. By using a regression, you can use one variable to predict what the other should be.

Finally, we used the residual or error from the regression to see how far off each prediction was. Just like we did with the original scores, we used standard deviation to find usually suspiciously large errors. Even though yes, our regression might not be perfect, times when it's very wrong probably call for an investigation.

# Discussion points#

• Why would this analysis be based on standard deviations away from the predicted value instead of just the predicted value?
• Standard deviation is how far away from the "average" a school is. Let's say you scored 3 standard deviations away from the average, but it was only a 5-point difference. What kind of situation could lead to that? Is it as important as being 3 standard deviations away but with a 50-point difference?
• The Dallas Morning News specifically called out schools with scores "more than three standard deviations away from what would be predicted based on their scores in other grades or on other tests." Do you think they ignored schools that were 2.99 standard deviations away?
• Did we ignore those schools? If we did, how could we be more cautious in the future?
• What are the pros and cons of selecting a cutoff like three standard deviations away from the predicted value? Note that three standard deviations is a typical number in stats
• What's the difference between a school with predicted scores -3 standard deviations away as compared to +3 standard deviations away? Do we need to pay attention to both, or only one?
• What next steps should we take after we've calculated these findings?
• If a school did have a strong math department and a weak English department, they would definitely be predicted incorrectly. What happens to that school after being flagged by research like this?
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