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Có thể tính phương sai và độ lệch chuẩn của số liệu ở biểu đồ trên hay không?

🔴Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Có thể tính phương sai và độ lệch chuẩn của số liệu ở biểu đồ trên hay không?

Phương pháp giải - Xem chi tiết

Phương sai của mẫu số liệu ghép nhóm, kí hiệu \({S^2}\), được tính bởi công thức: \({S^2} = \frac{1}{n}[{n_1}{({c_1} - \overline x )^2} + {n_2}{({c_2} - \overline x )^2} + ... + {n_k}{({c_k} - \overline x )^2}]\) Trong đó: \(n = {n_1} + {n_2} + ... + {n_k}\) là cỡ mẫu \(\overline x  = \frac{1}{n}({n_1}{c_1} + {n_2}{c_2} + ... + {n_k}{c_k})\) là số trung bình Độ lệch chuẩn của mẫu số liệu ghép nhóm, kí hiệu \(S\), là căn bậc hai số học của phương sai.

Lời giải chi tiết

Cỡ mẫu: n = 21. Giá trị trung bình của mẫu số liệu mới: \(\overline x  = \frac{1}{n}({n_1}{c_1} + {n_2}{c_2} + ... + {n_k}{c_k}) \) \(= \frac{1}{{21}}(3.162 + 5.166 + 8.170 + 4.174 + 1.178)\) \(= \frac{{3550}}{{21}}\). Phương sai của mẫu số liệu mới: \({S^2} = \frac{1}{n}[{n_1}{({c_1} - \overline x )^2} + {n_2}{({c_2} - \overline x )^2} + ... + {n_k}{({c_k} - \overline x )^2}] \) \(= \frac{1}{{21}}[3{(162 - \frac{{3550}}{{21}})^2} + 5{(166 - \frac{{3550}}{{21}})^2} + ... + 1{(178 - \frac{{3550}}{{21}})^2}]\) \(= \frac{{8000}}{{441}}\). Độ lệch chuẩn của mẫu số liệu mới: \(\sigma  = \sqrt {{S^2}}  = \sqrt {\frac{{8000}}{{441}}}  = \frac{{40\sqrt 5 }}{{21}}\).

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