Mean, Median and Mode: Formula, Null Hypothesis, Interpretation, SPSS, R, Python and Excel Guide

What Are Mean, Median and Mode?

Mean is the arithmetic average. It is calculated by adding all values and dividing by the number of observations. Median is the middle value after sorting the data from smallest to largest. Mode is the most frequently occurring value. These three statistics are called measures of central tendency because they describe the center of a dataset.

In a perfectly balanced distribution, the mean, median and mode are often close to each other. When the distribution is skewed or contains outliers, the three measures may separate. The mean is sensitive to extreme values, the median is more resistant to outliers, and the mode shows the most common observed value. That is why students should learn mean median and mode together rather than memorizing each one separately.

In the G3 final-grade example, the mean = 11.91, median = 12.00 and mode = 11. This means the average final grade is about 12, the middle student also has a score around 12, and the most common single score is 11. The three values tell a consistent story of a central grade range around 11 to 12, but the low-score tail means we should still review spread, skewness and the frequency distribution.

Mean, Median and Mode Formula

Mean Formula

Mean = Σx / n

In this formula, Σx means the sum of all values and n means the number of observations. For G3, the SPSS output reports a mean of 11.91, meaning the average final grade is just under 12.

Median Formula

If n is odd: Median = value at position (n + 1) / 2
If n is even: Median = average of the two middle values

The median is found after arranging the values in order. For G3, the median is 12.00. This means half of the valid observations are at or below the middle position and half are at or above the middle position.

Mode Formula

Mode = value with the highest frequency

The mode is the most repeated value. In the G3 frequency table, score 11 appears 104 times, or 16.0% of valid cases. Therefore, the mode is 11.

Null and Alternative Hypothesis for Mean, Median and Mode

Mean, median and mode are usually descriptive statistics. They do not automatically produce a p-value like a one-sample z test or a t test. Still, a hypothesis-style statement is useful when we are using central tendency to judge whether a distribution has one clear center or whether skewness/outliers make the center harder to summarize.

For G3, the mean is 11.91, median is 12.00 and mode is 11. The mean-median gap is only about 0.09, which is very small. The mode is one grade point lower than the median, but that is not unusual for an integer grade variable. Therefore, the descriptive decision is to fail to reject the null hypothesis that G3 has a common central location around 12.

Dataset and Clean SPSS-Ready Files Used

This example uses the student-por.csv student performance dataset. The main outcome variable is G3 final grade. Additional numeric variables were also checked so that the post can show how mean and median behave across variables with different shapes, including G1, G2, age, absences, studytime and failures.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified SPSS, R and Python Results

The SPSS output, Python charts and R charts agree on the main G3 central tendency result. The G3 final grade has mean = 11.91, median = 12.00 and mode = 11. The frequency table shows that score 11 is the most common G3 score, while the histogram and boxplot show that most values are concentrated around 10 to 15 with a small number of very low values.

Main G3 Central Tendency Result

Hypothesis Result for Central Tendency Alignment

Central Tendency Across Numeric Variables

G3 Frequency Table Summary

Python Charts and Interpretation

1. Mean, Median and Mode for G3

This histogram places the three central tendency measures directly on the G3 distribution. The dashed mean line is at about 11.91, the solid median line is at 12.00, and the dotted mode line is at 11.00. The bars show that most students scored between about 10 and 15, while a smaller number of students scored very low values such as 0 or 1. The mean and median lines almost overlap, which supports the descriptive null hypothesis that G3 has a common center around 12. The mode line is slightly to the left because score 11 is the most repeated exact score. This does not contradict the mean or median; it simply tells us that the single most frequent grade is 11 even though the overall center is around 12. The low-score values on the left side explain why the mean is slightly below the median and why SPSS reports negative skewness. Therefore, the chart supports a clear central-tendency conclusion while also reminding the reader that outliers and skewness should be discussed.

2. Central Tendency Comparison for G3

This bar chart separates the three central tendency values so they can be compared quickly. The mean bar is 11.91, the median bar is 12.00, and the mode bar is 11.00. The visual difference between mean and median is very small, which indicates that the average and the middle position give almost the same central answer. The mode is lower by one point because 11 has the highest frequency. In a discrete grade variable, the mode often differs slightly from the mean and median because it depends on one exact value rather than the full distribution. For hypothesis wording, this chart supports failing to reject the descriptive null hypothesis that the main center is consistent. The practical conclusion is that G3 is centered near 12, with 11 as the most common observed grade.

3. Mean vs Median Across Numeric Variables

This chart compares mean and median for several numeric variables. Large gaps between the two measures suggest skewness, outliers or a bounded scale. For G3, the mean and median are very close, which supports a stable central interpretation. For G2 and G1, the mean and median are also close, suggesting that grade variables have reasonably similar central locations. The biggest contrast appears for absences, where the mean is about 3.66 and the median is 2. This large gap means absences are right-skewed: many students have low absences, but some students have high absence counts that pull the mean upward. The chart also shows why the median is often more informative for skewed count variables. For age, mean and median are close because most students are around 16 to 17. Overall, this chart teaches that mean and median should be read together. When they agree, the center is easy to summarize. When they separate, the analyst should check skewness, outliers, a histogram or a boxplot.

4. Boxplots for Central Tendency and Spread

The boxplot chart explains why mean, median and mode should not be interpreted without spread. Each box shows the middle 50% of values, the line inside the box shows the median, and the marker shows the mean. For G3, the median is around 12, the box covers a moderate grade range, and several low outliers appear near 0 and 1. These low outliers explain the negative skewness and why the mean is slightly below the median. For G1 and G2, the boxes are similar but slightly lower, showing earlier grade periods with centers around 11. For age, the box is narrow because most students are in a small age range, with a few older values. For absences, the box is close to the lower end, but many high outliers extend upward, confirming strong right skew. For studytime, values are concentrated in low categories. This chart supports the conclusion that G3 has a useful central value around 12, but it also shows why central tendency must be accompanied by spread and outlier interpretation.

5. Frequency Distribution of G3

This frequency distribution shows the exact count for each G3 score. The tallest bar occurs at G3 = 11, which is why the mode is 11. The next very common values are around 10, 13, 12 and 14, showing that the central grade region is dense. The distribution also includes 15 students with G3 = 0 and only 2 students with G3 = 19. These rare edge values are important because they affect the mean and shape of the distribution. The mode is especially useful here because it identifies the most common exact score, while the median identifies the middle position and the mean uses every score. The chart supports the final interpretation that the typical G3 score is around 11 to 12, with 11 being the most frequent single value.

R Validation Charts

The R charts reproduce the same central tendency interpretation using a separate software workflow. This is useful because it confirms that the result is not limited to Python or SPSS. The R charts support the same conclusion: G3 has mean ≈ 11.91, median = 12 and mode = 11.

The R histogram validates the same pattern seen in Python. The mean and median reference lines are close to each other around 12, while the mode line is slightly lower at 11. Most observations are grouped in the central grade range, but a small low-score tail remains visible. This confirms the descriptive hypothesis decision: the central tendency values are close enough to describe a common center, but distribution shape still needs a short qualification.

The R bar chart confirms the values mean = 11.91, median = 12 and mode = 11. The main teaching point is that the mean and median are almost identical, so the central location is stable. The mode is still useful because it identifies the most common exact grade, not the arithmetic center. In a final report, this chart can be used as a simple visual summary of the three central tendency measures.

The R mean-versus-median chart confirms that G3, G1, G2 and age have relatively close mean and median values, while absences shows a much larger gap. This is exactly what the SPSS statistics show: absences has strong positive skewness, while G3 has a central mean-median alignment with a separate low-score tail. This chart is important because it prevents overgeneralization. A dataset can contain some variables with stable centers and other variables with skewed centers.

The R boxplot chart validates the spread and outlier interpretation. G3 has a central box around the middle grade range and a few very low outliers. Absences has a low median and many high outliers, which explains why its mean is much higher than its median. The chart shows that central tendency must be interpreted with distribution spread. Mean, median and mode answer “where is the center,” while boxplots answer “how spread out and unusual are the values around that center.”

The R frequency distribution confirms that the modal value of G3 is 11. The tallest bar marks the most common score, while neighboring bars around 10 to 14 show the dense central grade region. This chart is the clearest visual support for the mode because the mode is defined by frequency, not by arithmetic calculation. It also supports the interpretation that the most typical exact score is 11, even though the overall center is around 12.

How to Calculate Mean, Median and Mode in Python, R, SPSS and Excel

Mean, Median and Mode in Python

Python can calculate mean, median, mode, frequency counts, skewness and kurtosis using pandas and scipy.

import pandas as pd
from scipy import stats

df = pd.read_csv("spss_ready_data.csv")
g3 = pd.to_numeric(df["G3"], errors="coerce").dropna()

n = g3.count()
mean_g3 = g3.mean()
median_g3 = g3.median()
mode_g3 = g3.mode().iloc[0]
sd_g3 = g3.std(ddof=1)
skew_g3 = stats.skew(g3, bias=False)
kurt_g3 = stats.kurtosis(g3, fisher=True, bias=False)

print("Null hypothesis: mean, median and mode are close enough to describe one common center.")
print("Alternative hypothesis: mean, median and mode differ enough to suggest skewness or outlier influence.")
print("N:", n)
print("Mean:", round(mean_g3, 2))
print("Median:", round(median_g3, 2))
print("Mode:", mode_g3)
print("SD:", round(sd_g3, 3))
print("Skewness:", round(skew_g3, 3))
print("Excess kurtosis:", round(kurt_g3, 3))

freq = g3.value_counts().sort_index()
print(freq)

Mean, Median and Mode in R

R calculates mean and median directly. The mode can be calculated with a small custom function because base R uses the word “mode” differently.

student <- read.csv("spss_ready_data.csv")
g3 <- na.omit(as.numeric(student$G3))

get_mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}

n <- length(g3)
mean_g3 <- mean(g3)
median_g3 <- median(g3)
mode_g3 <- get_mode(g3)
sd_g3 <- sd(g3)

cat("Null hypothesis: mean, median and mode are close enough to describe one common center.\n")
cat("Alternative hypothesis: central tendency values differ enough to suggest skewness or outlier influence.\n")
cat("N:", n, "\n")
cat("Mean:", round(mean_g3, 2), "\n")
cat("Median:", round(median_g3, 2), "\n")
cat("Mode:", mode_g3, "\n")
cat("SD:", round(sd_g3, 3), "\n")

print(table(g3))

Mean, Median and Mode in SPSS

SPSS can calculate mean, median and mode through the FREQUENCIES command. The syntax below also exports the SPSS output to PDF.

* Mean Median and Mode Analysis in SPSS.
* Descriptive hypothesis:
* H0: mean, median and mode are close enough to describe one common center.
* H1: central tendency measures differ enough to suggest skewness or outlier influence.

SET UNICODE=ON.
SET DECIMAL=DOT.
SET PRINTBACK=OFF.
SET MPRINT=OFF.

GET DATA
 /TYPE=TXT
 /FILE="D:\DATA ANALYSIS\A Basic Descriptive Statistics Guides\Mean Median and Mode\SPSS_Output\clean_data\spss_ready_data.csv"
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /IMPORTCASE=ALL.

DATASET NAME StudentData WINDOW=FRONT.

TITLE "Mean Median and Mode Analysis".

FREQUENCIES VARIABLES=G3 G2 G1 age absences studytime failures
 /STATISTICS=MEAN MEDIAN MODE STDDEV VARIANCE SKEWNESS SESKEW KURTOSIS SEKURT RANGE MINIMUM MAXIMUM
 /ORDER=ANALYSIS.

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

EXAMINE VARIABLES=G3 BY school
 /STATISTICS DESCRIPTIVES
 /CINTERVAL 95
 /MISSING LISTWISE
 /NOTOTAL.

EXAMINE VARIABLES=G3 BY sex
 /STATISTICS DESCRIPTIVES
 /CINTERVAL 95
 /MISSING LISTWISE
 /NOTOTAL.

OUTPUT EXPORT
 /CONTENTS EXPORT=VISIBLE
 /PDF DOCUMENTFILE="D:\DATA ANALYSIS\A Basic Descriptive Statistics Guides\Mean Median and Mode\SPSS_Output\Mean-Median-Mode-SPSS-Output.pdf".

Mean, Median and Mode in Excel

Excel has direct formulas for mean, median and mode. The formulas below assume the G3 values are in cells B2:B650.

How to Report Mean, Median and Mode

A strong central tendency report should include the sample size, mean, median, mode, standard deviation, range, skewness and a short visual interpretation. Do not report the mean alone when the distribution has outliers or skewness. For G3, the mean and median agree closely, but the mode and frequency distribution add important detail.

If the goal is to compare groups or test a formal mean difference, use the correct inferential test instead of only mean, median and mode. For example, compare assumptions with Levene test, Brown-Forsythe test or Cochran C test. For repeated-measures assumptions, see Mauchly’s test of sphericity and Greenhouse-Geisser correction. For model diagnostics, see Ramsey RESET test and Goldfeld-Quandt test.

Related Data Analysis Guides

Mean, median and mode are part of basic descriptive statistics. Continue with Descriptive Statistics, Frequency Distribution, Five Number Summary, Histogram Interpretation, Box Plot Interpretation, Coefficient of Variation, Confidence Interval, Effect Size, Central Limit Theorem, Q-Q Plot Normality Check and P-P Plot Normality Check.

Common Mistakes

1. Reporting only the mean

The mean is useful, but it can be affected by outliers. In this G3 example, the mean and median are close, but the low-score tail still matters. Always check the median, mode and distribution shape.

2. Thinking the mode is always the best “typical” value

The mode is the most frequent value, not necessarily the best overall center. For G3, the mode is 11, but the median is 12 and the mean is 11.91. The typical grade is better described as around 11 to 12.

3. Ignoring skewness

Mean and median can look close even when outliers exist. SPSS reports G3 skewness as -0.913, so the low-score tail should be mentioned.

4. Forgetting that mode can be multimodal

Some datasets have more than one mode. If two or more values are tied for the highest frequency, the distribution is multimodal. In this G3 output, score 11 is the single modal value.

5. Using mean for ordinal categories without caution

Variables such as studytime, health and famrel are ordered categories. Their means can be useful for summaries, but medians and frequency distributions are often easier to interpret.

6. Treating descriptive hypotheses as formal p-value tests

The null and alternative hypothesis in this post are descriptive framing tools. They help explain central tendency alignment. A formal hypothesis test requires a specific inferential method and a test statistic.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the frequency tables, mean, median, mode, standard deviation, skewness, kurtosis, histograms, boxplots and group descriptives used in this guide.

External References for Mean, Median and Mode

This post uses verified Python, R and SPSS outputs together with standard statistical documentation and dataset references.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• pandas documentation
• Microsoft Excel support documentation