def pooled_variance(distribution1, distribution2, verbose=False):
"""
Get the pooled variance of the distribution, where the sample sizes are not similar.
Parameters
----------
> distribution1: an array of integers containing the distribution values of the first sample
> distribution2: an array of integers containing the distribution values of the second sample
> verbose (optional): a boolean that prints means and sum of squares of the samples before returning pooled variance if `True`;
print nothing if `False`
Returns
-------
The pooled variance of the samples
"""
xbar1 = get_mean(distribution1)
squares1 = [(xi - xbar1)**2 for xi in distribution1]
ssx = sum(squares1)
xbar2 = get_mean(distribution2)
squares2 = [(xi - xbar2)**2 for xi in distribution2]
ssy = sum(squares2)
n1 = len(distribution1)
n2 = len(distribution2)
if verbose:
print(f"Mean of sample 1: {xbar1}")
print(f"Sum of squares for sample 1: {ssx}")
print(f"Mean of sample 2: {xbar2}")
print(f"Sum of squares for sample 2: {ssy}")
return (ssx + ssy) / (get_dof(n1) + get_dof(n2))
def get_y_intercept(x_dist, y_dist, r): """ y = mx + c => c = y - mx = ybar - r(sy/sx)xbar """ ybar = get_mean(y_dist) xbar = get_mean(x_dist) sy = bessel_correction(y_dist)['Sample SD'] sx = bessel_correction(x_dist)['Sample SD'] m = get_slope(r, sy, sx) return ybar - m * xbar
def get_dependent_stats(x1, x2):
"""
Get the following statistics for two dependent distributions.
1. First sample and sample mean
2. Second sample and sample mean
3. Difference of both the samples and difference mean
Parameters
----------
> x1: an array containing elements of the first distribution
> x2: an array containing elements of the second distribution
Returns a dictionary with the following key-value pairs.
{
'first_sample': x1, # the array `x1`
'first_sample_mean': mean1, # mean of the first sample
'second_sample': x2, # the array `x2`
'second_sample_mean': mean2, # mean of the second sample
'difference': D, # an array containing the differences between each corresponding element of `x1` and `x2`
'mean_difference': mean_diff # mean of the above `difference` array
}
"""
D = []
l = len(x1)
for i in range(l):
D.append(x2[i] - x1[i])
mean1 = get_mean(x1)
mean2 = get_mean(x2)
mean_diff = get_mean(D)
return ({
"first_sample": x1,
"first_sample_mean": mean1,
"second_sample": x2,
"second_sample_mean": mean2,
"difference": D,
"mean_difference": mean_diff
})
def sum_squared_between(samples): """ Get the sum of squares for between-group variability of the samples. Parameter --------- > `samples`: a tuple of lists, where each list is a sample containing all the values of that sample Returns ------- The sum of squares for between-group variability. """ xbarG = get_grand_mean(samples) # grand mean ss = 0 # sum of squares for between-group variability for sample in samples: xbarK = get_mean(sample) n = len(sample) ss += n * ((xbarK - xbarG) ** 2) return ss
def honestly_significant_samples(samples, q_critical, verbose=True):
"""
Get / print the honestly significant samples among the tuple of samples.
Assumption: All samples have the same size.
Parameters
----------
> `samples`: a tuple of lists, where each list is a sample containing all the values of that sample
> `q_critical`: The Studentized Range Statistic at a certain alpha level
> `verbose`: a `bool` that governs whether or not the indices of significantly different samples be printed (defaulted to `True`)
Returns
-------
A list tuples where each tuple contains a pair of honestly significant means.
"""
ms_with = ms_within(samples)
n = len(samples[0])
# all samples must have the same size
k = len(samples)
for i in range(1, k):
if not len(samples[i]) == n:
raise "Samples do not have the same size"
THSD = tukey_HSD(q_critical, ms_with, n) # Tukey's HSD
means = [get_mean(sample) for sample in samples]
significantly_different_means = []
for i in range(k - 1):
m1 = means[i]
for j in range(i+1, k):
m2 = means[j]
diff = m1 - m2
if diff < 0: diff = -1 * diff # difference should always be +ve
if diff > THSD:
significantly_different_means.append((m1, m2))
if verbose:
print(f"Means of samples indexed {i} and {j} are honestly significantly different")
return significantly_different_means
# t-tests
## t-statistic
xbar = 6.47
s = 0.4
n = 500
mu0 = 6.07
t = get_t_stat(xbar, mu0, None, s, n)
print(f"t-statistic for these parameters is {t}")
print()
males = [41, 56, 82, 39, 3, 55, 70, 32, 46, 28, 39, 38, 47, 44, 45, 43, 28, 43, 56, 56, 33, 68, 49, 17, 40, 2, 28, 35, 27, 39, 46, 33, 30, 72, 28, 52, 47, 50, 25, 39]
famles = [93, 40, 36, 62, 52, 59, 59, 37, 58, 45, 33, 43, 32, 37, 51, 84, 30, 72, 63, 42, 60, 30, 29 ,52, 58, 50, 56, 42]
SE = 4.01
t = get_t_stat(get_mean(males), get_mean(famles), SE)
print(f"t for quiz = {t}")
print()
## t-critical
alpha = 0.05
dof = 12
n = 30
t_critical = get_t_critical(get_dof(n), alpha, tails=2)
print(f"t-critical value for alpha level {alpha} and sample size {n} = {t_critical}")
print()
## t-test
if t_test(t, t_critical):