For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $φ$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle φ(λ_{1}) μ(\{x \in Ω: f(x) \geq λ_{1} \}) + \sum_{k=2}^{n}\left(φ(λ_{k})- φ(λ_{k-1})\right) μ(\{x \in Ω: f(x) \geq λ_{k}\}) ,$ where $0 < λ_1 < λ_2 \cdots λ_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.