Quick wins for a faster PC:
Clear out junk files and repair common Windows errorsFree Scan →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Use scipy.stats.chisquare to compare observed category counts with specified expected frequencies; use scipy.stats.chi2_contingency to test whether categorical variables are independent in a contingency table. Both return a statistic and p-value. The second also returns degrees of freedom and expected cell counts.
Choose the function that matches your question
| Question | SciPy function | Input | Expected counts |
|---|---|---|---|
| Do counts for one categorical variable differ from stated proportions? | scipy.stats.chisquare |
Observed frequencies and, usually, expected frequencies in matching categories | You provide them; if omitted, SciPy assumes equal category probabilities. |
| Are two or more categorical variables independent? | scipy.stats.chi2_contingency |
A table of observed frequencies | SciPy calculates them from the table margins under the independence assumption. |
These are distinct tests, even though both use the chi-square distribution for their usual asymptotic p-values. The SciPy chisquare reference describes a goodness-of-fit test; the chi2_contingency reference describes the contingency-table test.
Run a goodness-of-fit test with chisquare
Use this test when you have one categorical variable and a specified distribution to compare against. Supply counts—not raw continuous measurements—and align observed and expected values category by category.
import numpy as np
from scipy.stats import chisquare
observed = np.array([16, 18, 16, 14, 12, 12])
expected = np.array([16, 16, 16, 16, 16, 8])
result = chisquare(f_obs=observed, f_exp=expected)
print(result.statistic, result.pvalue)Here, f_obs contains the observed frequencies, and f_exp contains the expected frequencies for those same categories. If you omit f_exp, SciPy treats the categories as equally likely. For expected proportions, convert them to expected counts using the sample total. For example, if the expected proportions are 0.5 and 0.5 and you observed 100 cases, the expected counts are 50 and 50.
The Tool Desk
Outbyte PC Repair FREERepair Windows errors before they cause bigger problemsFix Now →Outbyte Driver Updater FREEScan for outdated or missing drivers - takes under a minuteDriver Scan →#1 Best Overall
- Read Before You Buy — No Video Output: These adapters support charging and USB 2.0 data transfer, but cannot transmit video signals. Except for standard USB webcams (which use USB data only), they are not compatible with HDMI/DisplayPort cables, video-capable USB-C hubs, or docking stations with video output.
- Convert USB-A Ports to USB-C: Designed to connect USB-C earphones, cables, flash drives, card readers, and other USB-C accessories to standard USB-A ports. Plug-and-play with no drivers or software required.
- Aluminum Alloy Housing: Built with a sturdy aluminum alloy shell that aids in heat dissipation and protects against daily wear and scratches. Designed to maintain a stable and secure connection.
- Compact & Travel-Friendly: The ultra-compact design allows the adapter to stay plugged into your device without blocking adjacent ports or adding bulk, reducing wear and tear on your original USB ports.
- 12-Month Warranty: Backed by a 12-month manufacturer warranty for peace of mind. Designed to meet strict quality control standards for reliable everyday performance.
Check totals and degrees of freedom
For the ordinary Pearson goodness-of-fit test, observed and expected frequencies should have matching totals. SciPy’s sum_check checks this by default; do not bypass that check unless the statistical model and the reason for doing so are clear.
The usual degrees of freedom are the number of categories minus one. If you estimated parameters from the data, the degrees of freedom may need to be reduced. SciPy documents k - 1 - p for the efficient maximum-likelihood case, where k is the number of categories and p is the number of estimated parameters. The function’s ddof argument adjusts the degrees of freedom used for the p-value. Parameter estimation can also mean that the asymptotic distribution is not chi-square, so a degrees-of-freedom adjustment is not automatically sufficient.
Rank #2
- 5-in-1 USB-C Hub: Experience comprehensive connectivity featuring a Power Delivery input, two USB-A 2.0 ports, a USB-A 3.0 port, and an HDMI port. (Note: The USB-C power delivery input port is only for connecting an external wall charger to power your laptop and cannot power peripheral devices.)
- 90W Pass-Through Charging: Achieve optimal charging with 90W pass-through power to your laptop, supported by a total input of 100W, with the hub reserving 10W for operational efficiency. (Note: Wall charger not included.)
- Quick Data Transfers: Accelerate your productivity with rapid data transfers using a high-speed 5Gbps USB 3.0 port and two 480Mbps USB 2.0 ports.
- 4K HDMI Display: Enhance your visual experience with a hub capable of delivering 4K resolution at 30Hz in both mirror and extend modes. Please note that this hub is compatible with MacBook (macOS 12 and newer), Windows 10 and 11, ChromeOS, and laptops equipped with DP Alt Mode and Power Delivery. Note: This device is not compatible with Linux.
- What You Get: Anker USB-C Hub (5-in-1, 4K HDMI), welcome guide, 18-month warranty, and our friendly customer service.
Test independence with chi2_contingency
Use this function for a cross-tabulation: rows and columns represent category values of different variables, and each cell holds the observed count. SciPy derives the counts expected under independence from the row and column totals.
import numpy as np
from scipy.stats import chi2_contingency
observed_table = np.array([
[10, 10, 20],
[20, 20, 20]
])
result = chi2_contingency(observed_table)
print("Statistic:", result.statistic)
print("p-value:", result.pvalue)
print("Degrees of freedom:", result.dof)
print("Expected frequencies:")
print(result.expected_freq)The result includes the test statistic, p-value, degrees of freedom, and expected-frequency table. Inspect the expected table before interpreting the asymptotic p-value, especially when some categories are uncommon.
Rank #3
- Sleek 7-in-1 USB-C Hub: Features an HDMI port, two USB-A 3.0 ports, and a USB-C data port, each providing 5Gbps transfer speeds. It also includes a USB-C PD input port for charging up to 100W and dual SD and TF card slots, all in a compact design.
- Flawless 4K@60Hz Video with HDMI: Delivers exceptional clarity and smoothness with its 4K@60Hz HDMI port, making it ideal for high-definition presentations and entertainment. (Note: Only the HDMI port supports video projection; the USB-C port is for data transfer only.)
- Double Up on Efficiency: The two USB-A 3.0 ports and a USB-C port support a fast 5Gbps data rate, significantly boosting your transfer speeds and improving productivity.
- Fast and Reliable 85W Charging: Offers high-capacity, speedy charging for laptops up to 85W, so you spend less time tethered to an outlet and more time being productive.
- What You Get: Anker USB-C Hub (7-in-1), welcome guide, 18-month warranty, and our friendly customer service.
Correction, statistic, and resampling options
- Continuity correction:
correction=Trueis the default. When the degrees of freedom equal one, SciPy applies Yates’ correction by moving each observed count 0.5 toward its expected count. Setcorrection=Falseif you intentionally want the uncorrected statistic. - Alternative statistics: The default
lambda_=Noneselects Pearson’s chi-square statistic. Settinglambda_selects another statistic from the Cressie–Read power-divergence family. - Resampling p-values: In the SciPy 1.18.0 documentation, the
methodoption supports permutation or Monte Carlo p-values only for a two-way table, withcorrection=Falseand the default lambda. The documented Monte Carlo setup usesscipy.stats.random_table. These constraints are version-sensitive; check the reference for the SciPy version you have installed.
Check whether the chi-square approximation is suitable
Both functions are intended for frequency counts in categorical data. A common diagnostic guideline is that observed and expected cell frequencies should be at least 5, but this is not a universal guarantee. SciPy warns that small observed or expected counts can make the approximation unreliable. Review the expected frequencies and the study design rather than treating one cutoff as proof that the test is valid.
If the counts are too sparse for a reliable approximation, choose an alternative that fits the design and table. SciPy’s related references include Fisher’s exact test for 2-by-2 tables and exact alternatives such as Barnard’s test; they are not interchangeable in every setting.
Rank #4
- Dual Converters, Infinite Potential：Includes 2× USB C male to USB A female adapters and 2× USB A male to USB C female adapters. Perfect for a wide range of uses—tablets with Bluetooth keyboards, expand USB ports on macbook, and more. Two different converters for all your daily needs
- Next-Level 10Gbps & 3A Charging: No more slow 480Mbps, this usb to usb c adapter has a transfer speed of up to 10Gbps, allowing you to do more transferring in less time. This usb adapter fits both USB A and USB C charger, supporting up to 3A fast charging
- Upgraded Exquisite Craftsmanship: With an aluminum alloy housing and metal connector, the usbc to usb adapter is extremely durable and sturdy. Rigorously tested to withstand more than 10,000 times of plugging and unplugging, ensuring long-lasting performance
- Broad Compatible: The usb c to usb adapter widely supports all USB C/ USB A devices like laptops, tablets, cellphones, car chargers, and phone chargers. Such as compatible with MacBook Pro/Air 2023/2022, Thunderbolt 4/3 Devices,Apple MagSafe Watch 9/8/7/SE/Ultra, iPad Pro 2022/2021, Samsung Galaxy S23/S20/S10, and iPhone 17/16/15 Pro. Plug and play
- Please Note: To reach 10Gbps speed, keep the cable under 3.3 ft. For USB A Male to USB C adapters, try flipping the USB C connector. USB C Male to USB A adapters support bidirectional 10Gbps transfer within 3.3 ft
Interpret the statistic and p-value correctly
The p-value measures how incompatible the observed counts are with the test’s null hypothesis, assuming the test’s conditions hold. A small p-value is evidence against that null; it does not give the probability that the null is true.
For an independence test, a small p-value does not identify which cells drive the result, show the direction of a relationship, or measure its practical strength. The test is two-sided. Consider examining the observed and expected table and reporting an association measure such as Cramer’s V when a measure of strength is useful. See SciPy’s chi-square hypothesis-testing tutorial and association measure reference.
Quick Recap
Best Value
- 5-in-1 Connectivity: Equipped with a 4K HDMI port, a 5 Gbps USB-C data port, two 5 Gbps USB-A ports, and a USB C 100W PD-IN port. Note: The USB C 100W PD-IN port supports only charging and does not support data transfer devices such as headphones or speakers.
- Powerful Pass-Through Charging: Supports up to 85W pass-through charging so you can power up your laptop while you use the hub. Note: Pass-through charging requires a charger (not included). Note: To achieve full power for iPad, we recommend using a 45W wall charger.
- Transfer Files in Seconds: Move files to and from your laptop at speeds of up to 5 Gbps via the USB-C and USB-A data ports. Note: The USB C 5Gbps Data port does not support video output.
- HD Display: Connect to the HDMI port to stream or mirror content to an external monitor in resolutions of up to 4K@30Hz. Note: The USB-C ports do not support video output.
- What You Get: Anker 332 USB-C Hub (5-in-1), welcome guide, our worry-free 18-month warranty, and friendly customer service.
Report enough detail for someone to understand the test
- Name the test and the variables or categories being analyzed.
- Show the observed counts or provide a clearly identified contingency table.
- Report the statistic, degrees of freedom, and p-value.
- For goodness-of-fit, state the expected proportions or counts and whether model parameters were estimated.
- For independence, include or summarize the expected-count check.
- State whether you used Yates’ correction, a non-default statistic, or resampling.
- Add an effect-size measure when useful; do not present the p-value as a measure of effect size or direction.
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
