Barry weights 20 oranges and 25 lemons.?

i)
Mean weight of the 45 fruits
= (Total weight) / 45
= (220 × 20 + 118 × 25) / 45 g
= 163.33 g

ii)
Variance, σ² = Σ(xᵢ - μ)²/n
σ² = Σ(xᵢ² - 2μxᵢ + μ²)/n
σ² = (Σxᵢ² - 2μΣxᵢ + nμ²)/n
σ² = (Σxᵢ²/n) - 2μ(Σxᵢ/n) + μ²
σ² = (Σxᵢ²/n) - μ²

For oranges:
32² = (Σxₒ²/20) - 220²
Σxₒ² = 988480 g²

For lemons:
12² = (Σxₗ²/25) - 118²
Σxₗ² = 351700

Variance of the weights of the 45 fruits, σ²
= (Σxᵢ²/n) - μ²
= [(Σxₒ² +Σxₗ²)/n] - μ²
= [(988480 + 351700)/45] - 163.33²
= 3105

220 = (O[1] + O[2] + ... + O[20]) / 20
4400 = O[1] + O[2] + ... + O[20] = o

118 = (L[1] + L[2] + ... + L[25]) / 25
118 * 25 = L[1] + L[2] + ... + L[25]
2950 = L[1] + L[2] + ... + L[25] = l

(O[1] + O[2] + ... + O[20] + L[1] + L[2] + ... + L[25]) / (25 + 20) =>
(4400 + 2950) / 45 =>
7350 / 45 =>
7 * 105 * 10 / (5 * 9) =>
7 * 21 * 10 / 9 =>
7 * 7 * 10 / 3 =>
490 / 3 =>
163.333333.....

Now let's look at those standard deviations

sqrt(((220 - O[1])^2 + (220 - O[2])^2 + ... + (220 - O[20])^2) / 20) = 32
((220 - O[1])^2 + (220 - O[2])^2 + ... + (220 - O[20])^2) / 20 = 1024
220^2 - 440 * O[1] + O[1]^2 + 220^2 - 440 * O[2] + O[2]^2 + ... + 220^2 - 440 * O[20] + O[2]^2 = 1024
220^2 * 20 - 440 * (O[1] + O[2] + ... + O[20]) + O[1]^2 + ... + O[20]^2 = 1024
20 * 48400 - 440 * o + O[1]^2 + O[2]^2 + ... + O[20]^2 = 1024
968000 - 440 * 4400 + O[1]^2 + O[2]^2 + ... + O[20]^2 = 1024
O[1]^2 + O[2]^2 + ... + O[20]^2 = 1024 + 968000
O[1]^2 + O[2]^2 + ... + O[20]^2 = 969024

sqrt(((118 - L[1])^2 + (118 - L[2])^2 + ... + (118 - L[25])^2) / 25) = 12
((118 - L[1])^2 + (118 - L[2])^2 + ... + (118 - L[25])^2) / 25 = 144
(118 - L[1])^2 + (118 - L[2])^2 + ... + (118 - L[25])^2 = 3600
118^2 - 236 * L[1] + L[1]^2 + ... + 118^2 - 236 * L[25] + L[25]^2 = 3600
25 * 118^2 - 236 * (L[1] + L[2] + ... + L[25]) + (L[1]^2 + L[2]^2 + ... + L[25]^2) = 3600
25 * 118^2 - 236 * 2950 + L[1]^2 + L[2]^2 + ... + L[25]^2 = 3600
L[1]^2 + L[2]^2 + .. + L[25]^2 = 351700


(m - O[1])^2 + (m - O[2])^2 + ... + (m - O[20])^2 + (m - L[1])^2 + (m - L[2])^2 + ... + (m - L[25])^2 =>
m^2 - 2m * O[1] + O[1]^2 + m^2 - 2 * m * O[2] + O[2]^2 + ... + m^2 - 2m * O[20] + O[20]^2 + m^2 - 2m * L[1] + L[1]^2 + m^2 - 2m * L[2] + L[2]^2 + ... + m^2 - 2m * L[25] + L[25]^2 =>
45 * m^2 - 2m * (O[1] + O[2] + ... + O[20] + L[1] + L[2] + ... + L[25]) + O[1]^2 + O[2]^2 + ... + O[20]^2 + L[1]^2 + L[2]^2 + ... + L[25]^2 =>
45 * (490/3)^2 - 2 * (490/3) * (4400 + 2950) + 969024 + 351700 =>
45 * (1/9) * 490^2 - (980/3) * 7350 + 969024 + 351700 =>
5 * 490^2 - 980 * 2450 + 1320724 =>
120224

Divide that by 45

120224 / 45 =>
2,671.6444444444444444444444444444

Take the square root

51.687952604494255054167555825693

To the nearest integer

52

52 grams is the standard deviation.

I don't know how to get 20 for this part because that just doesn't make any sense.  The mean is 163.333... grams and the average weights for each group is 118 and 220, with rather small standard deviations for each.  It'd make sense that the standard deviation would be 52, since that's the interval where we'd expect to find 68% of the data set.