# Example Chi Square test - why different answers from HypothesisTests and Distances?
**URL:** https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392
**Category:** Statistics
**Tags:** first-steps
**Created:** [August 13, 2018, 4:10pm UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392 "2018-08-13T16:10:33Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@robblackwell](https://discourse.julialang.org/u/robblackwell)
#### Post date: [August 13, 2018, 4:10pm UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/1 "2018-08-13T16:10:34Z")
I’m trying to reproduce an example Chi square test given in in the text book, Practical Statistics for Field Biology which compares the histogram of some numbers (O) with expected values (E). The worked example gives 5.2 with 9 degrees of freedom.
`HypothesisTests` gives 5.2 but says 19 degrees of freedom, `Distances` gives the answer 2.74
I’m wondering whether a statistician would be kind enough to comment on the below code and let me know what I’m doing wrong please. Ultimately, I’m trying to learn how to compare the histograms of two grey-scale images, and give a p-value indicating their similarity or otherwise.
```julia
julia> using Distances
julia> O = [10 7 10 6 14 8 11 11 12 11]
1×10 Array{Int64,2}:
10 7 10 6 14 8 11 11 12 11
julia> E = [10 10 10 10 10 10 10 10 10 10]
1×10 Array{Int64,2}:
10 10 10 10 10 10 10 10 10 10
julia> chisq_dist(E,O)
2.7429759782700955
julia> using HypothesisTests
julia> ChisqTest(hcat(E,O))
Pearson's Chi-square Test
-------------------------
Population details:
parameter of interest: Multinomial Probabilities
value under h_0: [0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05]
point estimate: [0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.035, 0.05, 0.03, 0.07, 0.04, 0.055, 0.055, 0.06, 0.055]
95% confidence interval: Tuple{Float64,Float64}[(0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.0, 0.0826), (0.005, 0.0976), (0.0, 0.0776), (0.025, 0.1176), (0.0, 0.0876), (0.01, 0.1026), (0.01, 0.1026), (0.015, 0.1076), (0.01, 0.1026)]
Test summary:
outcome with 95% confidence: fail to reject h_0
one-sided p-value: 0.9992
Details:
Sample size: 200
statistic: 5.200000000000001
degrees of freedom: 19
residuals: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.948683, 0.0, -1.26491, 1.26491, -0.632456, 0.316228, 0.316228, 0.632456, 0.316228]
std. residuals: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.973329, 0.0, -1.29777, 1.29777, -0.648886, 0.324443, 0.324443, 0.648886, 0.324443]
```
---
### Author:  [@alejandromerchan](https://discourse.julialang.org/u/alejandromerchan)
#### Post date: [August 13, 2018, 5:20pm UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/2 "2018-08-13T17:20:59Z")
I think you need to clarify what Chi-Square test do you want to perform, goodness of fit or a contingency table?
If you run your own code to estimate the goodness of fit
```julia
sum(((Obs-Exp).^2./Exp))
```
you obtain the 5.2 answer.
I can reproduce this answer by running
```julia
using HypothesisTests
ChisqTest(O)
```
with the correct degrees of freedom, 9.
By testing both O and E, you’re really running a long goodness of fit, where the deviations occur only in O, noe in E, obtaining the same Chi Square value, but different degrees of freedom.
I assume the code in Distances tests a contingency table too.
---
### Author:  [@robblackwell](https://discourse.julialang.org/u/robblackwell)
#### Post date: [August 14, 2018, 10:04am UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/3 "2018-08-14T10:04:57Z")
Thank you - Indeed, `HypothesisTests` and `Distances` do agree if using a contingency table:
```julia
julia> using HypothesisTests
julia> O = [10 7 10 6 14 8 11 11 12 11]
1×10 Array{Int64,2}:
10 7 10 6 14 8 11 11 12 11
julia> E = [10 10 10 10 10 10 10 10 10 10]
1×10 Array{Int64,2}:
10 10 10 10 10 10 10 10 10 10
julia> ChisqTest(vcat(E,O)')
Pearson's Chi-square Test
-------------------------
Population details:
parameter of interest: Multinomial Probabilities
value under h_0: [0.05, 0.0425, 0.05, 0.04, 0.06, 0.045, 0.0525, 0.0525, 0.055, 0.0525, 0.05, 0.0425, 0.05, 0.04, 0.06, 0.045, 0.0525, 0.0525, 0.055, 0.0525]
point estimate: [0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.035, 0.05, 0.03, 0.07, 0.04, 0.055, 0.055, 0.06, 0.055]
95% confidence interval: Tuple{Float64,Float64}[(0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.005, 0.0976), (0.0, 0.0826), (0.005, 0.0976), (0.0, 0.0776), (0.025, 0.1176), (0.0, 0.0876), (0.01, 0.1026), (0.01, 0.1026), (0.015, 0.1076), (0.01, 0.1026)]
Test summary:
outcome with 95% confidence: fail to reject h_0
one-sided p-value: 0.9736
Details:
Sample size: 200
statistic: 2.7429759782700973
degrees of freedom: 9
residuals: [0.0, 0.514496, 0.0, 0.707107, -0.57735, 0.333333, -0.154303, -0.154303, -0.301511, -0.154303, 0.0, -0.514496, 0.0, -0.707107, 0.57735, -0.333333, 0.154303, 0.154303, 0.301511, 0.154303]
std. residuals: [0.0, 0.760652, 0.0, 1.04257, -0.870388, 0.494166, -0.230663, -0.230663, -0.451985, -0.230663, 0.0, -0.760652, 0.0, -1.04257, 0.870388, -0.494166, 0.230663, 0.230663, 0.451985, 0.230663]
```
---
### Author:  [@robblackwell](https://discourse.julialang.org/u/robblackwell)
#### Post date: [November 28, 2019, 3:38pm UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/4 "2019-11-28T15:38:10Z")
For completeness, the goodness of fit test can be run like this:
```julia
julia> using HypothesisTests
julia> using LinearAlgebra
julia> O = [10, 7, 10, 6, 14, 8, 11, 11, 12, 11];
julia> E = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10];
julia> ChisqTest(O,normalize(E,1))
Pearson's Chi-square Test
-------------------------
Population details:
parameter of interest: Multinomial Probabilities
value under h_0: [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]
point estimate: [0.1, 0.07, 0.1, 0.06, 0.14, 0.08, 0.11, 0.11, 0.12, 0.11]
95% confidence interval: Tuple{Float64,Float64}[(0.02, 0.1837), (0.0, 0.1537), (0.02, 0.1837), (0.0, 0.1437), (0.06, 0.2237), (0.0, 0.1637), (0.03, 0.1937), (0.03, 0.1937), (0.04, 0.2037), (0.03, 0.1937)]
Test summary:
outcome with 95% confidence: fail to reject h_0
one-sided p-value: 0.8165
Details:
Sample size: 100
statistic: 5.200000000000001
degrees of freedom: 9
residuals: [0.0, -0.9486832980505138, 0.0, -1.2649110640673518, 1.2649110640673518, -0.6324555320336759, 0.31622776601683794, 0.31622776601683794, 0.6324555320336759, 0.31622776601683794]
std. residuals: [0.0, -1.0, 0.0, -1.3333333333333333, 1.3333333333333333, -0.6666666666666666, 0.3333333333333333, 0.3333333333333333, 0.6666666666666666, 0.3333333333333333]
```
---
### Author:  [@brett\_knoss](https://discourse.julialang.org/u/brett_knoss)
#### Post date: [February 26, 2020, 8:24am UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/5 "2020-02-26T08:24:43Z")
What if I want to use a column from a DataFrame? How would I define variables? I already am able to find the mean and median with describe().
---
### Author:  [@nalimilan](https://discourse.julialang.org/u/nalimilan)
#### Post date: [February 26, 2020, 9:12am UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/6 "2020-02-26T09:12:55Z")
See [FreqTables.jl](https://github.com/nalimilan/FreqTables.jl) to compute frequency tables.
---
### Author:  [@pdeffebach](https://discourse.julialang.org/u/pdeffebach)
#### Post date: [February 26, 2020, 1:43pm UTC](https://discourse.julialang.org/t/example-chi-square-test-why-different-answers-from-hypothesistests-and-distances/13392/7 "2020-02-26T13:43:32Z")
you can also pass your own functions to `describe`.