Outlier Detection: IQR, Z-Score, Modified Z-Score, Python, R and SPSS Guide

What Is Outlier Detection?

Outlier Detection is the process of finding values that are unusually low or unusually high compared with the rest of the data. In practical statistics, these values matter because they can affect the mean, standard deviation, regression models, normality checks, charts and many hypothesis tests.

An outlier may happen for several reasons. It may be a data-entry mistake, a measurement problem, a rare but valid observation, or a meaningful extreme case. For example, a student with 32 absences is unusual compared with most students, but that does not prove the value is false. It means the record should be checked before making final statistical decisions.

This guide uses three common outlier detection methods: the IQR rule, the z-score rule and the modified z-score rule. Boxplots are used to show the spread of each variable, bar charts compare the number of flagged cases, and scatterplots show how flagged absence records relate to final grade.

For related visual interpretation, review Box Plot Interpretation, Histogram Interpretation, Five Number Summary, Descriptive Statistics and Coefficient of Variation.

Outlier Detection Methods and Formulas

There is no single outlier detection method that works best for every dataset. The IQR rule is robust because it uses quartiles. The z-score rule is easy to understand but depends strongly on the mean and standard deviation. The modified z-score rule uses the median and median absolute deviation, so it can be more stable when data are skewed.

1. IQR Outlier Detection Rule

The interquartile range method uses the first quartile, third quartile and IQR. Values outside the lower and upper fences are flagged as potential outliers.

IQR = Q3 - Q1
Lower IQR fence = Q1 - 1.5 × IQR
Upper IQR fence = Q3 + 1.5 × IQR

Flag as potential outlier if:
x < lower IQR fence or x > upper IQR fence

For G3 final grade, SPSS reported Q1 = 10, Q3 = 14 and IQR = 4. Therefore, the lower IQR fence is 4 and the upper IQR fence is 20. Since the maximum G3 value is 19, the upper side does not create flagged G3 outliers. The low values below 4 are the main G3 outlier concern.

2. Z-Score Outlier Detection Rule

The z-score method converts each value into standard deviation units from the mean. In this guide, SPSS used the common rule |z| > 3.

z = (x - mean) / standard deviation

Flag as potential outlier if:
|z| > 3

This rule flagged 30 cases in at least one selected variable. It is useful when variables are roughly symmetric, but it can under-flag or over-flag values when a variable is highly skewed.

3. Modified Z-Score Outlier Detection Rule

The modified z-score method uses the median and median absolute deviation. It is often more robust than the ordinary z-score because the median is less affected by extreme values.

Modified z-score = 0.6745 × (x - median) / MAD

Flag as potential outlier if:
|Modified z-score| > 3.5

In this output, the modified z-score rule flagged the highest number of absence outliers. That is expected because absences are strongly right-skewed, with many low values and a smaller number of large absence counts.

Dataset and Variables Used

The outlier detection workflow used 649 valid cases and five selected numeric variables. The SPSS output reported 0 missing cases for all selected variables, so the listwise valid N was also 649.

When outlier detection is part of a broader modelling workflow, it should be combined with assumption checks such as Q-Q Plot Normality Check, P-P Plot Normality Check, Kolmogorov-Smirnov Test and DAgostino Pearson Test.

Null and Alternative Hypotheses for Outlier Rules

Outlier detection rules are usually decision rules rather than formal significance tests. Still, it is useful to state clear working hypotheses before reviewing the results. In this guide, the hypotheses are written as rule-based screening decisions, not p-value tests.

Verified SPSS, Python and R Results

The SPSS output, Python charts and R validation charts all show that different outlier detection methods can flag different numbers of cases. This is normal because the IQR rule, z-score rule and modified z-score rule define “unusual” in different ways.

SPSS Descriptive Output

The SPSS descriptive output confirmed that all selected variables had 649 valid cases and 0 missing cases. G3 final grade had a mean of 11.91 and standard deviation of 3.231. School absences had a mean of 3.66 and standard deviation of 4.641. Age ranged from 15 to 22. G1 and G2 both ranged from 0 to 19.

Z-Score Outlier Counts from SPSS

Comparison of IQR, Z-Score and Modified Z-Score Counts

The method comparison shows why it is risky to rely on only one outlier rule. For G3, all three methods flagged 16 cases. For absences, the IQR rule flagged 21 cases, the z-score rule flagged 11, and the modified z-score rule flagged 32. For age, the z-score rule flagged 3 cases, the IQR rule flagged 1, and the modified z-score rule flagged 0.

Extreme Value Review

SPSS listed the highest and lowest values for each variable. For G3 final grade, the highest values were 19 and 18, while the lowest listed values were 0. For school absences, the highest listed values were 32, 30, 26, 24 and 22. These absence values explain the strong right tail in the boxplots and scatterplot.

Python Charts and Interpretation

1. Boxplots for Numeric Variables

This chart compares the spread of G3, absences, age, G1 and G2 on the same value scale. The x-axis lists the variables and the y-axis shows their values. G3 has a median near 12 and an IQR roughly from 10 to 14, with low outlier points near 0 and 1. Absences has a much lower median near 2, but it has several high outliers above the upper whisker, reaching 32. Age is tightly grouped around 16 to 18, with a small number of older values above the whisker. G1 and G2 are centred around 11, but both show low-end outliers near 0. The main conclusion is that outliers are not evenly distributed across variables. Absences is the strongest right-skewed variable, while grade variables have important low-end extreme values.

2. Outlier Counts by Detection Method

This chart shows how many cases were flagged by each rule for each variable. The x-axis lists the variables and the y-axis shows the number of flagged cases. For G3, all three methods flag 16 cases, which means the low G3 values are unusual under all rules. For absences, the modified z-score method flags the most cases, followed by the IQR rule and then the ordinary z-score rule. This happens because absences are highly skewed and the median-based rule reacts differently from the mean-based z-score rule. For G1 and G2, the IQR rule is more sensitive than the z-score methods. The chart supports the hypothesis section because H0 is rejected for different numbers of cases depending on the rule used.

3. IQR Outlier Fences for G3

This histogram focuses on G3 final grade. The x-axis shows G3 scores and the y-axis shows frequency. The dashed vertical lines mark the lower and upper IQR fences. The lower fence is about 4 and the upper fence is about 20. Most G3 scores fall between 8 and 15, which is the main body of the distribution. Scores near 0 and 1 sit below the lower IQR fence and are therefore flagged as potential outliers. The upper fence is above the maximum observed G3 value of 19, so no upper-side G3 values are flagged by the IQR rule. In hypothesis terms, H0 is rejected for low G3 cases below the lower fence and not rejected for the main body of grades inside the fences.

4. Absences vs G3 with Flagged Outliers

This scatterplot places absences on the x-axis and G3 final grade on the y-axis. Non-flagged records appear as the regular group, while flagged outliers are highlighted. The chart shows many students clustered between 0 and 10 absences, with G3 scores spread mostly from about 7 to 17. Flagged cases appear in several contexts: very high absence counts, very low G3 scores, and some unusual combinations such as high absences with moderate or high grades. This is useful because it shows that outliers should be reviewed in context. A high absence count may be unusual, but the final grade can still vary widely.

R Validation Charts

The R charts validate the same outlier detection pattern outside Python. The visual style is different, but the main conclusions are the same: absences have the strongest high-end outlier pattern, G3 has low-end grade outliers, and the three detection methods flag different numbers of cases.

1. R Boxplots for Numeric Variables

The R boxplot confirms the same distribution patterns seen in Python. Absences show a compact lower range with many high-end points above the whisker. G3, G1 and G2 have central grade ranges around 10 to 14 but also show low-end values near 0. Age is the most compact variable, with only a small number of possible high values. This chart supports the conclusion that outlier handling should be variable-specific rather than applying one blanket decision to all records.

2. R Outlier Counts by Method

This R chart provides labelled counts, making the method comparison easy to verify. G3 has 16 flagged cases under all three methods. Absences show the biggest method difference: 21 under the IQR rule, 32 under the modified z-score rule and 11 under the z-score rule. G1 has 16 IQR flags but only 1 z-score and 1 modified z-score flag. G2 has 25 IQR flags and 7 flags under each z-score method. The chart confirms that the modified z-score is especially sensitive for the skewed absence variable, while the IQR rule is more sensitive for low grade values in G1 and G2.

3. R IQR Outlier Fences for G3

The R histogram again shows that the lower G3 fence sits near 4 and the upper fence near 20. The main G3 scores sit between the two fences, while the very low values near 0 and 1 fall outside the lower fence. Because G3 cannot exceed 19 in this dataset, the upper fence does not flag high-side grade values. This confirms that G3 outlier detection is mainly a low-performance or zero-grade review issue, not a high-grade issue.

4. R Scatterplot of Absences vs G3

The R scatterplot uses different point shapes to separate flagged and non-flagged cases. The x-axis shows absences and the y-axis shows G3 final grade. The flagged cases are spread across both low-grade and high-absence situations. Some flagged records have zero absences but very low grades, while others have high absences but moderate grades. This makes the practical lesson clear: outlier detection should not be interpreted without looking at the actual record context.

How to Run Outlier Detection in Python, R, SPSS and Excel

Outlier Detection in Python

Python can calculate IQR flags, z-score flags and modified z-score flags, then save charts for WordPress and clean files for SPSS review.

import numpy as np
import pandas as pd
from scipy import stats

numeric_vars = ["G3", "absences", "age", "G1", "G2"]

def iqr_flags(series):
    q1 = series.quantile(0.25)
    q3 = series.quantile(0.75)
    iqr = q3 - q1
    lower = q1 - 1.5 * iqr
    upper = q3 + 1.5 * iqr
    return (series < lower) | (series > upper), lower, upper

def modified_z_flags(series):
    median = series.median()
    mad = np.median(np.abs(series - median))
    if mad == 0:
        return pd.Series(False, index=series.index)
    modified_z = 0.6745 * (series - median) / mad
    return modified_z.abs() > 3.5

summary = []

for col in numeric_vars:
    x = df[col].dropna()
    iqr_out, lower, upper = iqr_flags(x)
    z = np.abs(stats.zscore(x, nan_policy="omit"))
    z_out = z > 3
    mz_out = modified_z_flags(x)

    summary.append({
        "variable": col,
        "iqr_outliers": int(iqr_out.sum()),
        "z_outliers_abs_gt_3": int(z_out.sum()),
        "modified_z_outliers_abs_gt_3_5": int(mz_out.sum()),
        "lower_iqr_fence": lower,
        "upper_iqr_fence": upper
    })

summary_table = pd.DataFrame(summary)
print(summary_table)

Outlier Detection in R

R can reproduce the same IQR, z-score and modified z-score decisions. This is useful when Python and R are used together for validation.

numeric_vars <- c("G3", "absences", "age", "G1", "G2")

iqr_flags <- function(x) {
  q1 <- quantile(x, 0.25, na.rm = TRUE)
  q3 <- quantile(x, 0.75, na.rm = TRUE)
  iqr <- q3 - q1
  lower <- q1 - 1.5 * iqr
  upper <- q3 + 1.5 * iqr
  x < lower | x > upper
}

modified_z_flags <- function(x) {
  med <- median(x, na.rm = TRUE)
  mad_value <- median(abs(x - med), na.rm = TRUE)
  if (mad_value == 0) return(rep(FALSE, length(x)))
  mz <- 0.6745 * (x - med) / mad_value
  abs(mz) > 3.5
}

outlier_summary <- data.frame()

for (v in numeric_vars) {
  x <- student[[v]]
  iqr_out <- iqr_flags(x)
  z_out <- abs(scale(x)) > 3
  mz_out <- modified_z_flags(x)

  outlier_summary <- rbind(
    outlier_summary,
    data.frame(
      variable = v,
      iqr_outliers = sum(iqr_out, na.rm = TRUE),
      z_outliers_abs_gt_3 = sum(z_out, na.rm = TRUE),
      modified_z_outliers_abs_gt_3_5 = sum(mz_out, na.rm = TRUE)
    )
  )
}

print(outlier_summary)

Outlier Detection in SPSS

SPSS can create standardized values and then flag cases where the absolute z-score is greater than 3.

* Descriptive statistics and Explore review.
DESCRIPTIVES VARIABLES=G3 absences age G1 G2
 /STATISTICS=MEAN STDDEV MIN MAX.

EXAMINE VARIABLES=G3 absences age G1 G2
 /PLOT BOXPLOT
 /STATISTICS DESCRIPTIVES EXTREME
 /CINTERVAL 95
 /MISSING LISTWISE
 /NOTOTAL.

* Save z-scores.
DESCRIPTIVES VARIABLES=G3 absences age G1 G2
 /SAVE
 /STATISTICS=MEAN STDDEV.

* Flag absolute z-scores greater than 3.
COMPUTE zflag_G3 = ABS(ZG3) > 3.
COMPUTE zflag_absences = ABS(Zabsences) > 3.
COMPUTE zflag_age = ABS(Zage) > 3.
COMPUTE zflag_G1 = ABS(ZG1) > 3.
COMPUTE zflag_G2 = ABS(ZG2) > 3.
COMPUTE any_outlier = MAX(zflag_G3,zflag_absences,zflag_age,zflag_G1,zflag_G2).
EXECUTE.

FREQUENCIES VARIABLES=zflag_G3 zflag_absences zflag_age zflag_G1 zflag_G2 any_outlier.

Outlier Detection in Excel

Excel can calculate the same screening rules using quartiles, standard deviation and median absolute deviation formulas.

How to Report Outlier Detection Results

A clear outlier detection report should name the rule, variables, sample size, cutoff values and number of flagged cases. It should also explain whether flagged cases were removed, retained or reviewed separately.

When outlier handling changes a result, report both versions if possible. For example, show the mean, standard deviation or model result with and without flagged cases. This is especially important for methods that rely on the mean and standard deviation. For follow-up summaries, see Confidence Interval, Effect Size and Frequency Distribution.

Related Data Analysis Guides

Outlier detection is closely connected with visual summaries and assumption checking. Start with Box Plot Interpretation, Five Number Summary, Descriptive Statistics and Histogram Interpretation. If outliers are affecting model assumptions, continue with Q-Q Plot Normality Check, P-P Plot Normality Check, Levene Test, Brown-Forsythe Test and Ramsey RESET Test.

Common Mistakes

1. Deleting every flagged outlier automatically

A flagged value is only a warning. It may be a real observation. Always check the case, variable meaning and possible data-entry issue before removing it.

2. Using only one outlier detection method

The IQR, z-score and modified z-score methods can flag different cases. In this example, absences had 11 z-score flags but 32 modified z-score flags. Method choice matters.

3. Ignoring skewness

School absences were strongly right-skewed. Skewed data often produce many upper-tail values. A median-based method can behave differently from a mean-based z-score method.

4. Treating outliers as proof of bad data

An outlier may be valid. A student with high absences or a very low grade may be unusual but still real. Statistical review should be paired with subject knowledge.

5. Reporting outlier removal without explanation

If cases are removed, explain the rule used, number removed and whether the conclusion changed. Silent outlier removal weakens trust in the analysis.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the descriptive statistics, case processing summary, boxplot and extreme value review, z-score outlier counts and listed flagged cases.

External References for Outlier Detection

This post uses verified Python, R and SPSS outputs together with standard statistical rules for IQR, z-scores and modified z-scores.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• SciPy statistical functions documentation
• Pandas documentation