diff --git a/jax/_src/numpy/reductions.py b/jax/_src/numpy/reductions.py
index a6f4e1f7b8e8..0eca75fbbb23 100644
--- a/jax/_src/numpy/reductions.py
+++ b/jax/_src/numpy/reductions.py
@@ -33,7 +33,7 @@
     _broadcast_to, check_arraylike, _complex_elem_type,
     promote_dtypes_inexact, promote_dtypes_numeric, _where, implements)
 from jax._src.lax import lax as lax_internal
-from jax._src.typing import Array, ArrayLike, DType, DTypeLike
+from jax._src.typing import Array, ArrayLike, DType, DTypeLike, DeprecatedArg
 from jax._src.util import (
     canonicalize_axis as _canonicalize_axis, maybe_named_axis,
     NumpyComplexWarning)
@@ -432,14 +432,22 @@ def _average(a: ArrayLike, axis: Axis = None, weights: ArrayLike | None = None,
 
 @implements(np.var, skip_params=['out'])
 def var(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
-        out: None = None, ddof: int = 0, keepdims: bool = False, *,
-        where: ArrayLike | None = None) -> Array:
-  return _var(a, _ensure_optional_axes(axis), dtype, out, ddof, keepdims,
+        out: None = None, correction: int | float = 0, keepdims: bool = False, *,
+        where: ArrayLike | None = None, ddof: int | DeprecatedArg = DeprecatedArg()) -> Array:
+  # TODO(vfdev-5): deprecated 2024-5-16, remove after deprecation expires.
+  if not isinstance(ddof, DeprecatedArg):
+    warnings.warn(
+      "The ddof argument of jax.numpy.var is deprecated and setting it "
+      "will soon raise an error. To avoid an error in the future, and to "
+      "suppress this warning, please use the correction argument instead.",
+      DeprecationWarning, stacklevel=2)
+    correction = ddof
+  return _var(a, _ensure_optional_axes(axis), dtype, out, correction, keepdims,
               where=where)
 
 @partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims'))
 def _var(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
-         out: None = None, ddof: int = 0, keepdims: bool = False, *,
+         out: None = None, correction: int | float = 0, keepdims: bool = False, *,
          where: ArrayLike | None = None) -> Array:
   check_arraylike("var", a)
   dtypes.check_user_dtype_supported(dtype, "var")
@@ -465,7 +473,7 @@ def _var(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
   else:
     normalizer = sum(_broadcast_to(where, np.shape(a)), axis,
                      dtype=computation_dtype, keepdims=keepdims)
-  normalizer = lax.sub(normalizer, lax.convert_element_type(ddof, computation_dtype))
+  normalizer = lax.sub(normalizer, lax.convert_element_type(correction, computation_dtype))
   result = sum(centered, axis, dtype=computation_dtype, keepdims=keepdims, where=where)
   return lax.div(result, normalizer).astype(dtype)
 
@@ -493,14 +501,22 @@ def _var_promote_types(a_dtype: DTypeLike, dtype: DTypeLike | None) -> tuple[DTy
 
 @implements(np.std, skip_params=['out'])
 def std(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
-        out: None = None, ddof: int = 0, keepdims: bool = False, *,
-        where: ArrayLike | None = None) -> Array:
-  return _std(a, _ensure_optional_axes(axis), dtype, out, ddof, keepdims,
+        out: None = None, correction: int | float = 0, keepdims: bool = False, *,
+        where: ArrayLike | None = None, ddof: int | DeprecatedArg = DeprecatedArg()) -> Array:
+  # TODO(vfdev-5): deprecated 2024-5-16, remove after deprecation expires.
+  if not isinstance(ddof, DeprecatedArg):
+    warnings.warn(
+      "The ddof argument of jax.numpy.var is deprecated and setting it "
+      "will soon raise an error. To avoid an error in the future, and to "
+      "suppress this warning, please use the correction argument instead.",
+      DeprecationWarning, stacklevel=2)
+    correction = ddof
+  return _std(a, _ensure_optional_axes(axis), dtype, out, correction, keepdims,
               where=where)
 
 @partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims'))
 def _std(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
-         out: None = None, ddof: int = 0, keepdims: bool = False, *,
+         out: None = None, correction: int | float = 0, keepdims: bool = False, *,
          where: ArrayLike | None = None) -> Array:
   check_arraylike("std", a)
   dtypes.check_user_dtype_supported(dtype, "std")
@@ -508,7 +524,7 @@ def _std(a: ArrayLike, axis: Axis = None, dtype: DTypeLike | None = None,
     raise ValueError(f"dtype argument to jnp.std must be inexact; got {dtype}")
   if out is not None:
     raise NotImplementedError("The 'out' argument to jnp.std is not supported.")
-  return lax.sqrt(var(a, axis=axis, dtype=dtype, ddof=ddof, keepdims=keepdims, where=where))
+  return lax.sqrt(var(a, axis=axis, dtype=dtype, correction=correction, keepdims=keepdims, where=where))
 
 
 @implements(np.ptp, skip_params=['out'])
diff --git a/jax/experimental/array_api/_statistical_functions.py b/jax/experimental/array_api/_statistical_functions.py
index c34fb1fc3af4..25e60ca130f7 100644
--- a/jax/experimental/array_api/_statistical_functions.py
+++ b/jax/experimental/array_api/_statistical_functions.py
@@ -18,9 +18,9 @@
 # jnp.var
 def std(x, /, *, axis=None, correction=0.0, keepdims=False):
   """Calculates the standard deviation of the input array x."""
-  return jax.numpy.std(x, axis=axis, ddof=correction, keepdims=keepdims)
+  return jax.numpy.std(x, axis=axis, correction=correction, keepdims=keepdims)
 
 
 def var(x, /, *, axis=None, correction=0.0, keepdims=False):
   """Calculates the variance of the input array x."""
-  return jax.numpy.var(x, axis=axis, ddof=correction, keepdims=keepdims)
+  return jax.numpy.var(x, axis=axis, correction=correction, keepdims=keepdims)
diff --git a/tests/lax_numpy_reducers_test.py b/tests/lax_numpy_reducers_test.py
index 861b9014c589..bee4e239f370 100644
--- a/tests/lax_numpy_reducers_test.py
+++ b/tests/lax_numpy_reducers_test.py
@@ -540,14 +540,16 @@ def testAverage(self, shape, dtype, axis, weights_shape, returned, keepdims):
                           rtol=tol, atol=tol)
 
   @jtu.sample_product(
+    test_fns=[(np.var, jnp.var), (np.std, jnp.std)],
     shape=[(5,), (10, 5)],
     dtype=all_dtypes,
     out_dtype=inexact_dtypes,
     axis=[None, 0, -1],
-    ddof=[0, 1, 2],
+    correction=[0, 1, 2],
     keepdims=[False, True],
   )
-  def testVar(self, shape, dtype, out_dtype, axis, ddof, keepdims):
+  def testStdOrVar(self, test_fns, shape, dtype, out_dtype, axis, correction, keepdims):
+    np_fn, jnp_fn = test_fns
     rng = jtu.rand_default(self.rng())
     args_maker = self._GetArgsMaker(rng, [shape], [dtype])
     @jtu.ignore_warning(category=RuntimeWarning,
@@ -555,11 +557,11 @@ def testVar(self, shape, dtype, out_dtype, axis, ddof, keepdims):
     @jtu.ignore_warning(category=NumpyComplexWarning)
     def np_fun(x):
       # Numpy fails with bfloat16 inputs
-      out = np.var(x.astype(np.float32 if dtype == dtypes.bfloat16 else dtype),
+      out = np_fn(x.astype(np.float32 if dtype == dtypes.bfloat16 else dtype),
                    dtype=np.float32 if out_dtype == dtypes.bfloat16 else out_dtype,
-                   axis=axis, ddof=ddof, keepdims=keepdims)
+                   axis=axis, ddof=correction, keepdims=keepdims)
       return out.astype(out_dtype)
-    jnp_fun = partial(jnp.var, dtype=out_dtype, axis=axis, ddof=ddof, keepdims=keepdims)
+    jnp_fun = partial(jnp_fn, dtype=out_dtype, axis=axis, correction=correction, keepdims=keepdims)
     tol = jtu.tolerance(out_dtype, {np.float16: 1e-1, np.float32: 1e-3,
                                     np.float64: 1e-3, np.complex128: 1e-6})
     if (jnp.issubdtype(dtype, jnp.complexfloating) and