diff --git a/Cargo.toml b/Cargo.toml
index ee6b6ba2e94..7596ce2ec7e 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -79,6 +79,7 @@ rand_hc = { path = "rand_hc", version = "0.1" }
 [dev-dependencies]
 # This has a histogram implementation used for testing uniformity.
 average = "0.9.2"
+rand_pcg = { path = "rand_pcg", version = "0.1" }
 # Only for benches:
 rand_hc = { path = "rand_hc", version = "0.1" }
 rand_xoshiro = { path = "rand_xoshiro", version = "0.1" }
diff --git a/src/distributions/cauchy.rs b/src/distributions/cauchy.rs
index 9ffd5601ae6..3ec9fee9651 100644
--- a/src/distributions/cauchy.rs
+++ b/src/distributions/cauchy.rs
@@ -68,31 +68,24 @@ mod test {
 
     #[test]
     #[cfg(not(miri))] // Miri doesn't support transcendental functions
-    fn test_cauchy_median() {
+    fn test_cauchy_averages() {
+        // NOTE: given that the variance and mean are undefined,
+        // this test does not have any rigorous statistical meaning.
         let cauchy = Cauchy::new(10.0, 5.0);
-        let mut rng = ::test::rng(123);
+        let mut rng = crate::test::rng(123);
         let mut numbers: [f64; 1000] = [0.0; 1000];
+        let mut sum = 0.0;
         for i in 0..1000 {
             numbers[i] = cauchy.sample(&mut rng);
+            sum += numbers[i];
         }
         let median = median(&mut numbers);
         println!("Cauchy median: {}", median);
-        assert!((median - 10.0).abs() < 0.5); // not 100% certain, but probable enough
-    }
-
-    #[test]
-    #[cfg(not(miri))] // Miri doesn't support transcendental functions
-    fn test_cauchy_mean() {
-        let cauchy = Cauchy::new(10.0, 5.0);
-        let mut rng = ::test::rng(123);
-        let mut sum = 0.0;
-        for _ in 0..1000 {
-            sum += cauchy.sample(&mut rng);
-        }
+        assert!((median - 10.0).abs() < 0.4); // not 100% certain, but probable enough
         let mean = sum / 1000.0;
         println!("Cauchy mean: {}", mean);
         // for a Cauchy distribution the mean should not converge
-        assert!((mean - 10.0).abs() > 0.5); // not 100% certain, but probable enough
+        assert!((mean - 10.0).abs() > 0.4); // not 100% certain, but probable enough
     }
 
     #[test]
diff --git a/src/distributions/unit_circle.rs b/src/distributions/unit_circle.rs
index d078f182d6f..ddd3caf71df 100644
--- a/src/distributions/unit_circle.rs
+++ b/src/distributions/unit_circle.rs
@@ -84,10 +84,17 @@ mod tests {
 
     #[test]
     fn value_stability() {
-        let mut rng = ::test::rng(2);
-        let dist = UnitCircle::new();
-        assert_eq!(dist.sample(&mut rng), [0.968253135419177, 0.2499717299034683]);
-        assert_eq!(dist.sample(&mut rng), [-0.2083028196300198, 0.9780643819985387]);
-        assert_eq!(dist.sample(&mut rng), [0.4284767927057159, -0.9035527865667964]);
+        let mut rng = crate::test::rng(2);
+        let expected = [
+                [-0.9965658683520504, -0.08280380447614634],
+                [-0.9790853270389644, -0.20345004884984505],
+                [-0.8449189758898707, 0.5348943112253227],
+            ];
+        let samples = [
+                UnitCircle.sample(&mut rng),
+                UnitCircle.sample(&mut rng),
+                UnitCircle.sample(&mut rng),
+            ];
+        assert_eq!(samples, expected);
     }
 }
diff --git a/src/distributions/unit_sphere.rs b/src/distributions/unit_sphere.rs
index 4d895ed2359..3cd94d6dbcc 100644
--- a/src/distributions/unit_sphere.rs
+++ b/src/distributions/unit_sphere.rs
@@ -79,13 +79,17 @@ mod tests {
 
     #[test]
     fn value_stability() {
-        let mut rng = ::test::rng(2);
-        let dist = UnitSphereSurface::new();
-        assert_eq!(dist.sample(&mut rng),
-                   [0.9754391834492048, 0.12388255134190126, -0.18213048307455182]);
-        assert_eq!(dist.sample(&mut rng),
-                   [0.45487349593184284, 0.5619516852114866, -0.6908693119445342]);
-        assert_eq!(dist.sample(&mut rng),
-                   [-0.8374011645099787, 0.5296803977011814, 0.13489983679919715]);
+        let mut rng = crate::test::rng(2);
+        let expected = [
+                [0.03247542860231647, -0.7830477442152738, 0.6211131755296027],
+                [-0.09978440840914075, 0.9706650829833128, -0.21875184231323952],
+                [0.2735582468624679, 0.9435374242279655, -0.1868234852870203],
+            ];
+        let samples = [
+                UnitSphereSurface.sample(&mut rng),
+                UnitSphereSurface.sample(&mut rng),
+                UnitSphereSurface.sample(&mut rng),
+            ];
+        assert_eq!(samples, expected);
     }
 }
diff --git a/src/lib.rs b/src/lib.rs
index 975805418c4..bd2dcae72a5 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -561,10 +561,12 @@ mod test {
     use super::*;
     #[cfg(all(not(feature="std"), feature="alloc"))] use alloc::boxed::Box;
 
+    /// Construct a deterministic RNG with the given seed
     pub fn rng(seed: u64) -> impl RngCore {
-        // For tests, we want a statistically good, fast, reproducible RNG. We
-        // do not need cryptographic strength. ChaCha8 is a good choice.
-        rand_chacha::ChaCha8Rng::seed_from_u64(seed)
+        // For tests, we want a statistically good, fast, reproducible RNG.
+        // PCG32 will do fine, and will be easy to embed if we ever need to.
+        const INC: u64 = 11634580027462260723;
+        rand_pcg::Pcg32::new(seed, INC)
     }
 
     #[test]
diff --git a/src/seq/mod.rs b/src/seq/mod.rs
index e4cca760f47..f1302f20bef 100644
--- a/src/seq/mod.rs
+++ b/src/seq/mod.rs
@@ -613,8 +613,10 @@ mod test {
             counts[permutation] += 1;
         }
         for count in counts.iter() {
-            let err = *count - 10000i32 / 24;
-            assert!(-50 <= err && err <= 50);
+            // Binomial(10000, 1/24) with average 416.667
+            // Octave: binocdf(n, 10000, 1/24)
+            // 99.9% chance samples lie within this range:
+            assert!(352 <= *count && *count <= 483, "count: {}", count);
         }
     }